n. \end{cases}$$ The reader may find it instructive to compare with the calculations in homology, checking the correctness of the calculation by comparison with the universal coefficient theorem. We shall later use Poincar\'{e} duality to give a quick proof that the cohomology algebra $H^*(\bR P^n;\bZ_2)$ is a truncated polynomial algebra $\bZ_2[x]/(x^{n+1})$, where $\deg\,x=1$. That is, for $1\leq q\leq n$, the unique non-zero element of $H^q(\bR P^n;\bZ_2)$ is the $q$th power of $x$. This means that the elements are so tightly bound together that knowledge of the cohomological behavior of a map $f: \bR P^m\rtarr \bR P^n$ on cohomology in degree one determines its behavior on cohomology in all higher degrees. We assume that $m\geq 1$ and $n\geq 1$ to avoid triviality. \begin{prop} Let $f: \bR P^m\rtarr \bR P^n$ be a map such that $f_*: \pi_1(\bR P^m)\rtarr \pi_1(\bR P^n)$ is non-zero. Then $m\leq n$. \end{prop} \begin{proof} Since $\pi_1(\bR P^1)=\bZ$ and $\pi_1(\bR P^m)=\bZ_2$ if $m\geq 2$, the result is certainly true if $n=1$. Thus assume that $n>1$ and assume for a contradiction that $m>n$. By the naturality of the Hurewicz isomorphism, $f_*: H_1(\bR P^m;\bZ)\rtarr H_1(\bR P^n;\bZ)$ is non-zero. By our universal coefficient theorems, the same is true for mod $2$ homology and for mod $2$ cohomology. That is, if $x$ is the non-zero element of $H^1(\bR P^n;\bZ_2)$, then $f^*(x)$ is the non-zero element of $H^1(\bR P^m;\bZ_2)$. By the naturality of cup products $$ (f^*(x))^m = f^*(x^m).$$ However, the left side is non-zero in $H^m(\bR P^m;\bZ_2)$ and the right side is zero since $x^m=0$ by our assumption that $m>n$. The contradiction establishes the conclusion. \end{proof} We use this fact together with covering space theory to prove a celebrated result known as the Borsuk-Ulam theorem. A map $g: S^m\rtarr S^n$ is said to be antipodal\index{antipodal map} if it takes pairs of antipodal points to pairs of antipodal points. It then induces a map $f: \bR P^m\rtarr \bR P^n$ such that the following diagram commutes: $$\diagram S^m\rto^g \dto_{p_m} & S^n \dto^{p_n}\\ \bR P^m \rto_f & \bR P^n,\\ \enddiagram$$ where $p_m$ and $p_n$ are the canonical coverings. \begin{thm} If $m>n\geq 1$, then there exist no antipodal maps $S^m\rtarr S^n$. \end{thm} \begin{proof} Suppose given an antipodal map $g:S^m\rtarr S^n$. According to the proposition, $f_*:\pi_1(\bR P^m)\rtarr \pi_1(\bR P^n)$ is zero. According to the fundamental theorem of covering space theory, there is a map $\tilde{f}: \bR P^m\rtarr S^n$ such that $p_n\com\tilde{f}= f$. Let $s\in S^m$. Then $\tilde{f}(p_m(s))=\tilde{f}(p_m(-s))$ must be either $g(s)$ or $g(-s)$, since these are the only two points in $p_n^{-1}(f(p_m(s)))$. Thus either $t=s$ or $t=-s$ satisfies $\tilde{f}(p_m(t))=g(t)$. Therefore, by the fundamental theorem of covering space theory, the maps $\tilde{f}\com p_m$ and $g$ must be equal since they agree on a point. This is absurd: $\tilde{f}\com p_m$ takes antipodal points to the same point, while $g$ was assumed to be antipodal. \end{proof} \begin{thm}[Borsuk-Ulam]\index{Borsuk-Ulam theorem} For any continuous map $f: S^n\rtarr \bR^n$, there exists $x\in S^n$ such that $f(x)=f(-x)$. \end{thm} \begin{proof} Suppose for a contradiction that $f(x)\neq f(-x)$ for all $x$. We could then define a continuous antipodal map $g: S^n\rtarr S^{n-1}$ by letting $g(x)$ be the point at which the vector from $0$ through $f(x)-f(-x)$ intersects $S^{n-1}$. \end{proof} \section{Obstruction theory} We give an outline of one of the most striking features of cohomology: the cohomology groups of a space $X$ with coefficients in the homotopy groups of a space $Y$ control the construction of homotopy classes of maps $X\rtarr Y$. As a matter of motivation, this helps explain why one is interested in general coefficient groups. It also explains why the letter $\pi$ is so often used to denote coefficient groups. \begin{defn} Fix $n\geq 1$. A connected space $X$ is said to be $n$-simple\index{nsimple space@$n$-simple space} if $\pi _{1}(X)$ is Abelian and acts trivially on the homotopy groups $\pi _{q}(X)$ for $q\leq n$; $X$ is said to be simple\index{simple space} if it is $n$-simple for all $n$. \end{defn} Let $(X,A)$ be a relative CW complex with relative skeleta $X^n$ and let $Y$ be an $n$-simple space. The assumption on $Y$ has the effect that we need not worry about basepoints. Let $f: X^{n}\rtarr Y$ be a map. We ask when $f$ can be extended to a map $X^{n+1}\rtarr Y$ that restricts to the given map on $A$. If we compose the attaching maps $S^{n} \rightarrow X$ of cells of $X\setminus A$ with $f$, we obtain elements of $\pi_{n}(Y)$. These elements specify a well defined ``obstruction cocycle''\index{obstruction cocycle} \[ c_{f}\in C^{n+1}(X,A;{\pi}_{n}(Y)). \] Clearly, by considering extensions cell by cell, $f$ extends to $X^{n+1}$ if and only if $c_{f} = 0$. This is not a computable criterion. However, if we allow ourselves to modify $f$ a little, then we can refine the criterion to a cohomological one that often is computable. If $f$ and $f'$ are maps $X^{n} \rightarrow Y$ and $h$ is a homotopy rel $A$ of the restrictions of $f$ and $f'$ to $X^{n-1}$, then $f$, $f'$, and $h$ together define a map \[ h(f,f'): (X\times I)^{n} \rtarr Y. \] Applying $c_{h(f,f')}$ to cells $j\times I$, we obtain a ``deformation cochain''\index{deformation cochain} \[ d_{f,f',h}\in C^{n}(X,A;{\pi}_{n}(Y)) \] such that $\delta d_{f,f',h} = c_{f}-c_{f'}$. Moreover, given $f$ and $d$, there exists $f'$ that coincides with $f$ on $X^{n-1}$ and satisfies $d_{f,f'} = d$, where the constant homotopy $h$ is understood. This gives the following result. \begin{thm} For $f: X^{n}\rtarr Y$, the restriction of $f$ to $X^{n-1}$ extends to a map $X^{n+1}\rightarrow Y$ if and only if $[c_{f}]=0$ in $H^{n+1}(X,A;{\pi}_{n}(Y))$. \end{thm} It is natural to ask further when such extensions are unique up to homotopy, and a similar argument gives the answer. \begin{thm} Given maps $f, f': X^{n}\rightarrow Y$ and a homotopy rel $A$ of their restrictions to $X^{n-1}$, there is an obstruction class in $H^{n}(X,A;{\pi}_{n}(Y))$ that vanishes if and only if the restriction of the given homotopy to $X^{n-2}$ extends to a homotopy $f\simeq f'$ rel $A$. \end{thm} \vspace{.1in} \begin{center} PROBLEMS \end{center} The first few problems here are parallel to those at the end of Chapter 16. \begin{enumerate} \item Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\tilde{X}$ and let $A$ be an Abelian group. Using cellular or singular chains, show that $$C^*(X;A)\iso \Hom_{\bZ[\pi]}(C_*(\tilde{X}),A).$$ \item Show that there is an isomorphism $$H^*(K(\pi,1);A) \iso \Ext^*_{\bZ[\pi]}(\bZ,A).$$ When $A$ is a commutative ring, the Ext groups have algebraically defined products, constructed as follows. The evident isomorphism $\bZ\iso\bZ\ten\bZ$ is covered by a map of free $\bZ[\pi]$-resolutions $P \rtarr P\ten P$, where $\bZ[\pi]$ acts diagonally on tensor products, $\al(x\ten y) = \al x\ten \al y$. This chain map is unique up to chain homotopy. It induces a map of chain complexes $$\Hom_{\bZ[\pi]}(P,A)\ten \Hom_{\bZ[\pi]}(P,A) \rtarr \Hom_{\bZ[\pi]}(P,A)$$ and therefore an induced product on Ext$^*_{\bZ[\pi]}(\bZ,A)$. Convince yourself that the isomorphism above preserves products and explain the intuition (don't worry about technical exactitude). \item* Now use homological algebra to determine $H^*(\bR P^{\infty};\bZ_2)$ as a ring. \item Use the previous problem to deduce the ring structure on $H^*(\bR P^n;\bZ_2)$ for each $n\geq 1$. \item Let $p: Y\rtarr X$ be a covering space with finite fibers, say of cardinality $n$. Construct a ``transfer homomorphism''\index{transfer homomorphism} $t: H^*(Y;A)\rtarr H^*(X;A)$ and show that $t\com p^*: H^*(X;A)\rtarr H^*(X;A)$ is multiplication by $n$. \item Let $X$ and $Y$ be CW complexes. Show that the interchange map $$t: X\times Y\rtarr Y\times X$$ satisfies $t_*([i]\ten[j])=(-1)^{pq}[j]\ten[i]$ for a $p$-cell of $X$ and a $q$-cell of $Y$. Deduce that the cohomology ring $H^*(X)$ is commutative in the graded sense:\index{commutativity!graded} $$ x\cup y = (-1)^{pq}y\cup x \ \ \text{if}\ \ \text{deg}\,x=p\ \tand\ \text{deg}\,y=q.$$ \end{enumerate} An ``$H$-space''\index{Hspace@$H$-space} is a space $X$ with a basepoint $e$ and a product $\ph: X\times X\rtarr X$ such that the maps $\la: X\rtarr X$ and $\rh: X\rtarr X$ given by left and right multiplication by $e$ are each homotopic to the identity map. Note that $\la$ and $\rh$ specify a map $X\wed X\rtarr X$ that is homotopic to the codiagonal or folding map $\bigtriangledown$, which restricts to the identity on each wedge summand. The following two problems are optional review exercises. \begin{enumerate} \item[7.] If $e$ is a nondegenerate basepoint for $X$, then $\ph$ is homotopic to a product $\ph'$ such that left and right multiplication by $e$ under the product $\ph'$ are both identity maps. \item[8.] Show that the product on $\pi_1(X,e)$ induced by the based map $\ph': X\times X\rtarr X$ agrees with the multiplication given by composition of paths and that both products are commutative. \item[9.] For an $H$-space $X$, the following diagram is commutative: $$\diagram X\times X \dto_{\ph} \rrto^{\DE\times\DE} & & X\times X\times X\times X \rrto^{\id\times t\times \id} & & X\times X\times X\times X \dto^{\ph\times \ph} \\ X \xto[0,4]_{\DE} & & & & X\times X \enddiagram$$ (Check it: it is too trivial to write down.) Let $X$ be $(n-1)$-connected, $n\geq 2$, and let $x\in H^n(X)$. \begin{enumerate} \item[(a)] Show that $\ph^*(x) = x\ten 1 + 1\ten x$. \item[(b)] Show that $$(\DE\times\DE)^*(\id\times\, t\times \id)^*(\ph\times \ph)^*(x\ten x) =x^2\ten 1 +(1+(-1)^n)(x\ten x)+1\ten x^2.$$ \item[(c)] Prove that, if $n$ is even, then either $2(x\ten x)=0$ in $H^*(X\times X)$ or $x^2\neq 0$. Deduce that $S^{n}$ cannot be an $H$-space if $n$ is even. \end{enumerate} \end{enumerate} \chapter{Derivations of properties from the axioms} Returning to the axiomatic approach to cohomology, we assume given a theory on pairs of spaces and give some deductions from the axioms. This may be viewed as a dualized review of what we did in homology, and we generally omit the proofs. The only significant difference that we will encounter is in the computation of the cohomology of colimits. In a final section, we show the uniqueness of (ordinary) cohomology with coefficients in $\pi$. Prior to that section, we make no use of the dimension axiom in this chapter. A ``generalized cohomology theory'' \index{cohomology theory!generalized} $E^*$ is defined to be a system of functors $E^q(X,A)$ and natural transformations $\de:E^q(A)\rtarr E^{q+1}(X,A)$ that satisfy all of our axioms except for the dimension axiom. Similarly, we have the notion of a generalized cohomology theory on CW pairs, and the following result holds. \begin{thm} A cohomology theory $E^*$ on pairs of spaces determines and is determined by its restriction to a cohomology theory $E^*$ on pairs of CW complexes. \end{thm} \section{Reduced cohomology groups and their properties} For a based space $X$, we define the reduced cohomology\index{reduced cohomology} of $X$ to be $$\tilde{E}^q(X)=E^q(X,*).$$ There results a direct sum decomposition $$ E^*(X) \iso \tilde{E}^*(X)\oplus E^*(*)$$ that is natural with respect to based maps. For $*\in A\subset X$, the summand $E^*(*)$ maps isomorphically under the map $E^*(X)\rtarr E^*(A)$, and the exactness axiom implies that there is a reduced long exact sequence $$\cdots\rtarr \tilde{E}^{q-1}(A)\overto{\de} E^q(X,A) \rtarr \tilde{E}^q(X)\rtarr \tilde{E}^q(A) \rtarr \cdots.$$ The unreduced cohomology groups are recovered as the special cases $$ E^*(X)=\tilde{E}^*(X_+)$$ of reduced ones, and similarly for maps. Relative cohomology groups are also special cases of reduced ones. \begin{thm} For any cofibration\index{cofibration} $i: A\rtarr X$, the quotient map $q: (X,A)\rtarr (X/A,*)$ induces an isomorphism $$\tilde{E}^*(X/A)=E^*(X/A,*)\iso E^*(X,A).$$ \end{thm} We may replace any inclusion $i: A\rtarr X$ by the canonical cofibration $A\rtarr Mi$ and then apply the result just given to obtain an isomorphism $$ E^*(X,A)\iso \tilde{E}^*(Ci).$$ \begin{thm} For a nondegenerately based space $X$, there is a natural isomorphism $$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ \end{thm} \begin{cor} Let $*\in A\subset X$, where $i: A\rtarr X$ is a cofibration between nondegenerately based spaces. In the long exact sequence $$\cdots\rtarr \tilde{E}^{q-1}(A)\overto{\de} \tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr \tilde{E}^q(A)\rtarr \cdots $$ of the pair $(X,A)$, the connecting homomorphism $\de$ is the composite $$\tilde{E}^{q-1}(A)\overto{\SI} \tilde{E}^{q}(\SI A) \overto{\pa^*}\tilde{E}^q(X/A).$$ \end{cor} \begin{cor} For any $n$ and $q$, $$\tilde{E}^q(S^n)\iso \tilde{E}^{q-n}(*).$$ \end{cor} \section{Axioms for reduced cohomology} \begin{defn} A reduced cohomology theory\index{cohomology theory!reduced} $\tilde{E}^*$ consists of functors $\tilde{E}^q$ from the homotopy category of nondegenerately based spaces to the category of Abelian groups that satisfy the following axioms. \begin{itemize} \item EXACTNESS\index{exactness axiom}\ \ If $i: A\rtarr X$ is a cofibration, then the sequence $$\tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr \tilde{E}^q(A)$$ is exact. \item SUSPENSION\index{suspension axiom}\ \ For each integer $q$, there is a natural isomorphism $$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ \item ADDITIVITY\index{additivity axiom}\ \ If $X$ is the wedge of a set of nondegenerately based spaces $X_i$, then the inclusions $X_i\rtarr X$ induce an isomorphism $$\tilde{E}^*(X) \rtarr \textstyle{\prod}_i\, \tilde{E}^*(X_i).$$ \item WEAK EQUIVALENCE\index{weak equivalence axiom}\ \ If $f:X\rtarr Y$ is a weak equivalence, then $$f^*: \tilde{E}^*(Y)\rtarr \tilde{E}^*(X)$$ is an isomorphism. \end{itemize} \end{defn} The reduced form of the dimension axiom would read $$\tilde{H}^0(S^0)=\pi \ \ \tand \ \ \tilde{H}^q(S^0)=0\ \text{for}\ q\neq 0.$$ \begin{thm} A cohomology theory $E^*$ on pairs of spaces determines and is determined by a reduced cohomology theory $\tilde{E}^*$ on nondegenerately based spaces. \end{thm} \begin{defn} A reduced cohomology theory\index{cohomology theory!reduced} $\tilde{E}^*$ on based CW complexes consists of functors $\tilde{E}^q$ from the homotopy category of based CW complexes to the category of Abelian groups that satisfy the following axioms. \begin{itemize} \item EXACTNESS\index{exactness axiom}\ \ If $A$ is a subcomplex of $X$, then the sequence $$\tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr \tilde{E}^q(A)$$ is exact. \item SUSPENSION\index{suspension axiom}\ \ For each integer $q$, there is a natural isomorphism $$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ \item ADDITIVITY\index{additivity axiom}\ \ If $X$ is the wedge of a set of based CW complexes $X_i$, then the inclusions $X_i\rtarr X$ induce an isomorphism $$\tilde{E}^*(X) \rtarr \textstyle{\prod}_i\, \tilde{E}^*(X_i).$$ \end{itemize} \end{defn} \begin{thm} A reduced cohomology theory $\tilde{E}^*$ on nondegenerately based spaces determines and is determined by its restriction to a reduced cohomology theory on based CW complexes. \end{thm} \begin{thm} A cohomology theory $E^*$ on CW pairs determines and is determined by a reduced cohomology theory $\tilde{E}^*$ on based CW complexes. \end{thm} \section{Mayer-Vietoris sequences in cohomology} We have Mayer-Vietoris sequences in cohomology just like those in homology. The proofs are the same. Poincar\'{e} duality between the homology and cohomology of manifolds will be proved by an inductive comparison of homology and cohomology Mayer-Vietoris sequences. We record two preliminaries. \begin{prop} For a triple $(X,A,B)$, the following sequence is exact:\index{triple!exact sequence of} $$\cdots E^{q-1}(A,B) \overto{\de} E^q(X,A)\overto{j^*} E^q(X,B)\overto{i^*} E^{q}(A,B)\rtarr \cdots.$$ Here $i:(A,B)\rtarr (X,B)$ and $j:(X,B)\rtarr (X,A)$ are inclusions and $\de$ is the composite $$E^{q-1}(A,B)\rtarr E^{q-1}(A)\overto{\de} E^q(X,A).$$ \end{prop} Now let $(X;A,B)$ be an excisive triad and set $C=A\cap B$. \begin{lem} The map $$E^*(X,C) \rtarr E^*(A,C)\oplus E^*(B,C) $$ induced by the inclusions of $(A,C)$ and $(B,C)$ in $(X,C)$ is an isomorphism. \end{lem} \begin{thm}[Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence} Let $(X;A,B)$ be an excisive triad and set $C=A\cap B$. The following sequence is exact: $$\cdots \rtarr E^{q-1}(C)\overto{\DE^*} E^q(X) \overto{\ph^*} E^q(A)\oplus E^q(B)\overto{\ps^*} E^{q}(C)\rtarr \cdots.$$ Here, if $i: C\rtarr A$, $j: C\rtarr B$, $k: A\rtarr X$, and $\ell: B\rtarr X$ are the inclusions, then $$\ph^*(\ch)= (k^*(\ch),\ell^*(\ch))\ \ \tand\ \ \psi^*(\al,\be)=i^*(\al)-j^*(\be) $$ and $\DE^*$ is the composite $$E^{q-1}(C)\overto{\de} E^q(A,C)\iso E^q(X,B)\rtarr E^q(X).$$ \end{thm} For the relative version, let $X$ be contained in some ambient space $Y$. \begin{thm}[Relative Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence!relative} The following sequence is exact: $$\cdots\rtarr E^{q-1}(Y,C)\overto{\DE^*} E^q(Y,X)\overto{\ph^*} E^q(Y,A)\oplus E^q(Y,B) \overto{\ps^*} E^{q}(Y,C)\rtarr \cdots.$$ Here, if $i: (Y,C)\rtarr (Y,A)$, $j: (Y,C)\rtarr (Y,B)$, $k: (Y,A)\rtarr (Y,X)$, and $\ell: (Y,B)\rtarr (Y,X)$ are the inclusions, then $$ \ph^*(\ch)= (k^*(\ch),\ell^*(\ch)) \ \ \tand\ \ \psi^*(\al,\be)=i^*(\al)-j^*(\be)$$ and $\DE^*$ is the composite $$E^{q-1}(Y,C)\rtarr E^{q-1}(A,C)\iso E^{q-1}(X,B)\overto{\de} E^{q}(Y,X).$$ \end{thm} \begin{cor} The absolute and relative Mayer-Vietoris sequences are related by the following commutative diagram: $$\diagram E^{q-1}(C)\rto^{\DE^*} \dto_{\de} & E^{q}(X) \rto^(0.4){\ph^*} \dto^{\de} & E^{q}(A)\oplus E^{q}(B)\rto^(0.6){\ps^*} \dto^{\de+\de} & E^{q}(C) \dto^{\de} \\ E^q(Y,C)\rto_(0.46){\DE^*} & E^{q+1}(Y,X) \rto_(0.35){\ph^*} & E^{q+1}(Y,A)\oplus E^{q+1}(Y,B) \rto_(0.65){\ps^*} & E^{q+1}(Y,C).\\ \enddiagram$$ \end{cor} \section{Lim$^1$ and the cohomology of colimits} In this section, we let $X$ be the union of an expanding sequence of subspaces $X_i$, $i\geq 0$. We shall use the additivity and weak equivalence axioms and the Mayer-Vietoris sequence to explain how to compute $E^*(X)$. The answer is more subtle than in homology because, algebraically, limits are less well behaved than colimits: they are not exact functors from diagrams of Abelian groups to Abelian groups. Rather than go into the general theory, we simply display how the ``first right derived functor'' $\lim^{1}$\index{lima@$\lim^{1}$} of an inverse sequence of Abelian groups can be computed. \begin{lem} Let $f_i: A_{i+1}\rtarr A_{i}$, $i\geq 1$, be a sequence of homomorphisms of Abelian groups. Then there is an exact sequence $$ 0\rtarr \lim\,A_i\overto{\be} \textstyle{\prod}_i A_i\overto{\al} \textstyle{\prod}_i A_i\rtarr \lim^{1}A_i\rtarr 0,$$ where $\al$ is the difference of the identity map and the map with coordinates $f_i$ and $\be$ is the map whose projection to $A_i$ is the canonical map given by the definition of a limit. \end{lem} That is, we may as well define $\lim^{1}A_i$ to be the displayed cokernel. We then have the following result. \begin{thm}\index{colimit!cohomology of} For each $q$, there is a natural short exact sequence $$0 \rtarr {\lim}^{1}\,E^{q-1}(X_i)\rtarr E^q(X)\overto{\pi} \lim\,E^q(X_i) \rtarr 0,$$ where $\pi$ is induced by the inclusions $X_i\rtarr X$. \end{thm} \begin{proof} We use the notations and constructions in the proof that homology commutes with colimits and consider the excisive triad $(\tel\,X_i;A,B)$ with $C=A\cap B$ constructed there. By the additivity axiom, $$E^*(A)=\textstyle{\prod}_i\, E^*(X_{2i}),\ \ E^*(B) =\textstyle{\prod}_i\, E^*(X_{2i+1}),\ \tand\ E^*(C)=\textstyle{\prod}_i\, E^*(X_i).$$ We construct the following commutative diagram, whose top row is the cohomology Mayer-Vietoris sequence of the triad $(\tel\, X_i;A,B)$ and whose bottom row is an exact sequence of the sort displayed in the previous lemma. \begin{footnotesize} $$\diagram \cdots \rto & E^q(\tel X_i) \rto \dto_{\iso} & E^q(A)\oplus E^q(B) \rto \dto_{\iso} & E^q(C) \rto \dto^{\iso} & E^{q+1}(\tel\,X_i) \dto^{\iso} \rto & \cdots \\ \cdots \rto & E^q(X) \rto^{\be'} \dto_{\pi'} & \prod E^q(X_i) \rto^{\al'} \dto_{\prod(-1)^{i}} & \prod_i E^q(X_i) \dto^{\prod_i(-1)^{i}} \rto & E^{q+1}(X) \rto & \cdots \\ 0 \rto & \lim\,E^q(X_i) \rto^{\be} & \prod_i E^q(X_i) \rto^{\al} & \prod_i E^q(X_i) \rto & \lim^1 E^q(X_i) \rto & 0. \\ \enddiagram$$ \end{footnotesize} The commutativity of the bottom middle square is a comparison based on the sign used in the Mayer-Vietoris sequence. Here the map $\pi'$ differs by alternating signs from the canonical map $\pi$, but this does not affect the conclusion. A chase of the diagram implies the result. \end{proof} The $\lim^1$ ``error terms'' are a nuisance, and it is important to know when they vanish. We say that an inverse sequence $f_i: A_{i+1}\rtarr A_i$ satisfies the Mittag-Leffler condition\index{Mittag-Leffler condition} if, for each fixed $i$, there exists $j\geq i$ such that, for every $k>j$, the image of the composite $A_k\rtarr A_i$ is equal to the image of the composite $A_j\rtarr A_i$. For example, this holds if all but finitely many of the $f_i$ are epimorphisms or if the $A_i$ are all finite. As a matter of algebra, we have the following vanishing result. \begin{lem} If the inverse sequence $f_i: A_{i+1}\rtarr A_i$ satisfies the Mittag-Leffler condition, then $\lim^1\,A_i=0$. \end{lem} For example, for $qn$, and $\tilde{H}_n(M;\pi)=0$ if $M$ is connected and is not compact. \end{thm} We can use this together with Mayer-Vietoris sequences to construct $R$-fun\-da\-men\-tal classes at compact subspaces from $R$-orientations. To avoid trivialities, we tacitly assume that $n>0$. (The trivial case $n=0$ forced the use of reduced homology in the statement; where arguments use reduced homology below, it is only to ensure that what we write is correct in dimension zero.) \begin{thm} Let $K$ be a compact subset of $M$. Then, for any coefficient group $\pi$, $H_i(M,M-K;\pi)=0$ if $i>n$, and an $R$-orientation of $M$ determines an $R$-fundamental class of $M$ at $K$. In particular, if $M$ is compact, then an $R$-orientation of $M$ determines an $R$-fundamental class of $M$. \end{thm} \begin{proof} First assume that $K$ is contained in a coordinate chart $U\iso \bR^n$. By excision and exactness, we then have $$H_i(M,M-K;\pi)\iso H_i(U,U-K;\pi)\iso \tilde{H}_{i-1}(U-K;\pi).$$ Since $U-K$ is open in $U$, the vanishing theorem implies that $\tilde{H}_{i-1}(U-K;\pi)=0$ for $i>n$. In fact, a lemma used in the proof of the vanishing theorem will prove this directly. In this case, an $R$-fundamental class in $H_n(M,M-U)$ maps to an $R$-fundamental class in $H_n(M,M-K)$. A general compact subset $K$ of $M$ can be written as the union of finitely many compact subsets, each of which is contained in a coordinate chart. By induction, it suffices to prove the result for $K\cup L$ under the assumption that it holds for $K$, $L$, and $K\cap L$. With any coefficients, we have the Mayer-Vietoris sequence \begin{multline*} \cdots \rtarr H_{i+1}(M,M-K\cap L)\overto{\DE}H_i(M,M-K\cup L)\\ \overto{\ps}H_i(M,M-K)\oplus H_i(M,M-L)\overto{\ph} H_i(M,M-K\cap L) \rtarr \cdots. \end{multline*} The vanishing of $H_i(M,M-K\cup L;\pi)$ for $i>n$ follows directly. Now take $i=n$ and take coefficients in $R$. Then $\ps$ is a monomorphism. The $R$-fun\-da\-men\-tal classes $z_K\in H_n(M,M-K)$ and $z_L\in H_n(M,M-L)$ determined by a given $R$-orientation both map to the $R$-fundamental class $z_{K\cap L}\in H_n(M,M-K\cap L)$ determined by the given $R$-orientation. Therefore $$\ph(z_K,z_L) = z_{K\cap L}- z_{K\cap L} = 0$$ and there exists a unique $z_{K\cup L}\in H_n(M,M-K\cup L)$ such that $$ \ps(z_{K\cup L}) = (z_K,z_L).$$ Clearly $z_{K\cup L}$ is an $R$-fundamental class of $M$ at $K\cup L$. \end{proof} The vanishing theorem also implies the following dichotomy, which we have already noticed in our examples of explicit calculations. \begin{cor} Let $M$ be a connected compact $n$-manifold, $n>0$. Then either $M$ is not orientable and $H_n(M;\bZ)=0$ or $M$ is orientable\index{orientable} and the map $$H_n(M;\bZ) \rtarr H_n(M,M-x;\bZ)\iso \bZ$$ is an isomorphism for every $x\in M$. \end{cor} \begin{proof} Since $M-x$ is connected and not compact, $H_n(M-x;\pi)=0$ and thus $$H_n(M;\pi) \rtarr H_n(M,M-x;\pi)\iso \pi$$ is a monomorphism for all coefficient groups $\pi$. In particular, by the universal coefficient theorem, $$H_n(M;\bZ)\ten \bZ_q \rtarr H_n(M,M-x;\bZ)\ten \bZ_q\iso \bZ_q$$ is a monomorphism for all positive integers $q$. If $H_n(M;\bZ)\neq 0$, then $H_n(M;\bZ)\iso \bZ$ with generator mapped to some multiple of a generator of $H_n(M,M-x;\bZ)$. By the mod $q$ monomorphism, the coefficient must be $\pm 1$. \end{proof} As an aside, the corollary leads to a striking example of the failure of the naturality of the splitting in the universal coefficient theorem. Consider a connected, compact, non-orientable $n$-manifold $M$. Let $x\in M$ and write $M_x$ for the pair $(M,M-x)$. Since $M$ is $\bZ_2$-orientable, the middle vertical arrow in the following diagram is an isomorphism between copies of $\bZ_2$: $$\diagram 0 \rto & H_n(M)\ten \bZ_2 \rto \dto_0 & H_n(M;\bZ_2) \rto \dto^{\iso} & \Tor_1^{\bZ}(H_{n-1}(M),\bZ_2) \rto \dto^0 & 0 \\ 0 \rto & H_n(M_x)\ten \bZ_2 \rto & H_n(M_x;\bZ_2) \rto & \Tor_1^{\bZ}(H_{n-1}(M_x),\bZ_2) \rto & 0. \\ \enddiagram$$ Clearly $H_{n-1}(M,M-x)=0$, and the corollary gives that $H_n(M)=0$. Thus the left and right vertical arrows are zero. If the splittings of the rows were natural, this would imply that the middle vertical arrow is also zero. \section{The proof of the vanishing theorem} Let $M$ be an $n$-manifold, $n>0$. Take all homology groups with coefficients in a given Abelian group $\pi$ in this section. We must prove the intuitively obvious statement that $H_i(M)=0$ for $i>n$ and the much more subtle statement that $H_n(M)=0$ if $M$ is connected and is not compact. The last statement is perhaps the technical heart of our proof of the Poincar\'e duality theorem. We begin with the general observation that homology is ``compactly supported''\index{compactly supported homology} in the sense of the following result. \begin{lem} For any space $X$ and element $x\in H_q(X)$, there is a compact subspace $K$ of $X$ and an element $k\in H_q(K)$ that maps to $x$. \end{lem} \begin{proof} Let $\ga: Y\rtarr X$ be a CW approximation of $X$ and let $x=\ga_*(y)$. If $y$ is represented by a cycle $z\in C_q(Y)$, then $z$, as a finite linear combination of $q$-cells, is an element of $C_q(L)$ for some finite subcomplex $L$ of $Y$. Let $K=\ga(L)$ and let $k$ be the image of the homology class represented by $z$. Then $K$ is compact and $k$ maps to $x$. \end{proof} We need two lemmas about open subsets of $\bR^n$ to prove the vanishing theorem, the first of which is just a special case. \begin{lem} If $U$ is open in $\bR^n$, then $H_i(U)=0$ for $i\geq n$. \end{lem} \begin{proof} Let $s\in H_i(U)$, $i\geq n$. There is a compact subspace $K$ of $U$ and an element $k\in H_i(K)$ that maps to $s$. We may decompose $\bR^n$ as a CW complex whose $n$-cells are small $n$-cubes in such a way that there is a finite subcomplex $L$ of $\bR^n$ with $K\subset L\subset U$. (To be precise, use a cubical grid with small enough mesh.) For $i>0$, the connecting homomorphisms $\pa$ are isomorphisms in the commutative diagram $$\diagram H_{i+1}(\bR^n,L) \rto \dto_{\pa} & H_{i+1}(\bR^n,U) \dto^{\pa} \\ H_i(L) \rto & H_i(U).\\ \enddiagram$$ Since $(\bR^n,L)$ has no relative $q$-cells for $q > n$, the groups on the left are zero for $i\geq n$. Since $s$ is in the image of $H_i(L)$, $s=0$. \end{proof} \begin{lem} Let $U$ be open in $\bR^n$. Suppose that $t\in H_n(\bR^n,U)$ maps to zero in $H_n(\bR^n,\bR^n-x)$ for all $x\in \bR^n-U$. Then $t=0$. \end{lem} \begin{proof} We prove the equivalent statement that if $s\in \tilde{H}_{n-1}(U)$ maps to zero in $\tilde{H}_{n-1}(\bR^n-x)$ for all $x\in \bR^n-U$, then $s=0$. Choose a compact subspace $K$ of $U$ such that $s$ is in the image of $\tilde{H}_{n-1}(K)$. Then $K$ is contained in an open subset $V$ whose closure $\bar{V}$ is compact and contained in $U$, hence $s$ is the image of an element $r\in\tilde{H}_{n-1}(V)$. We claim that $r$ maps to zero in $\tilde{H}_{n-1}(U)$, so that $s=0$. Of course, $r$ maps to zero in $\tilde{H}_{n-1}(\bR^n-x)$ if $x\not\!\!{\in}\,U$. Let $T$ be an open contractible subset of $\bR^n$ such that $\bar{V}\subset T$ and $\bar{T}$ is compact. For example, $T$ could be a large enough open cube. Let $L=T-(T\cap U)$. For each $x\in \bar{L}$, choose a closed cube $D$ that contains $x$ and is disjoint from $V$. A finite set $\sset{D_1,\ldots\!,D_q}$ of these cubes covers $\bar{L}$. Let $C_i=D_i\cap T$ and observe that $(\bR^n-D_i)\cap T = T-C_i$. We see by induction on $p$ that $r$ maps to zero in $\tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_p))$ for $0\leq p\leq q$. This is clear if $p=0$. For the inductive step, observe that $$T-(C_1\cup\cdots\cup C_p) = (T-(C_1\cup\cdots\cup C_{p-1}))\cap (\bR^n-D_p)$$ and that $H_n((T-(C_1\cup\cdots\cup C_{p-1}))\cup (\bR^n-D_p)) = 0$ by the previous lemma. Therefore the map $$\tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_p))\rtarr \tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_{p-1}))\oplus \tilde{H}_{n-1}(\bR^n-D_p)$$ in the Mayer-Vietoris sequence is a monomorphism. Since $r\in\tilde{H}_{n-1}(V)$ maps to zero in the two right-hand terms, by the induction hypothesis and the contractibility of $D_p$ to a point $x\not\!\!{\in}\,U$, it maps to zero in the left-hand term. Since $$V\subset T-(C_1\cup\cdots\cup C_q)\subset T\cap U\subset U,$$ this implies our claim that $r$ maps to zero in $\tilde{H}_{n-1}(U)$. \end{proof} \begin{proof}[Proof of the vanishing theorem] Let $s\in H_i(M)$. We must prove that $s=0$ if $i>n$ and if $i=n$ when $M$ is connected and not compact. Choose a compact subspace $K$ of $M$ such that $s$ is in the image of $H_i(K)$. Then $K$ is contained in some finite union $U_1\cup\cdots\cup U_q$ of coordinate charts, and it suffices to prove that $H_i(U_1\cup\cdots\cup U_q)=0$ for the specified values of $i$. Inductively, using that $H_i(U)=0$ for $i\geq n$ when $U$ is an open subset of a coordinate chart, it suffices to prove that $H_i(U\cup V)=0$ for the specified values of $i$ when $U$ is a coordinate chart and $V$ is an open subspace of $M$ such that $H_i(V)=0$ for the specified values of $i$. We have the Mayer-Vietoris sequence $$H_i(U)\oplus H_i(V) \rtarr H_i(U\cup V) \rtarr \tilde{H}_{i-1}(U\cap V) \rtarr \tilde{H}_{i-1}(U)\oplus \tilde{H}_{i-1}(V).$$ If $i>n$, the vanishing of $H_i(U\cup V)$ follows immediately. Thus assume that $M$ is connected and not compact and consider the case $i=n$. We have $H_n(U)=0$, $H_n(V)=0$, and $\tilde{H}_{n-1}(U)=0$. It follows that $H_n(U\cup V)=0$ if and only if $i_*: \tilde{H}_{n-1}(U\cap V)\rtarr \tilde{H}_{n-1}(V)$ is a monomorphism, where $i: U\cap V\rtarr V$ is the inclusion. We claim first that $H_n(M)\rtarr H_n(M,M-y)$ is the zero homomorphism for any $y\in M$. If $x\in M$ and $L$ is a path in $M$ connecting $x$ to $y$, then the diagram $$\diagram & & H_n(M,M-x) \\ H_n(M) \rto & H_n(M,M-L) \urto^{\iso} \drto_{\iso} & \\ & & H_n(M,M-y) \\ \enddiagram$$ shows that if $s\in H_n(M)$ maps to zero in $H_n(M,M-x)$, then it maps to zero in $H_n(M,M-y)$. If $s$ is in the image of $H_n(K)$ where $K$ is compact, we may choose a point $x\in M-K$. Then the map $K\rtarr M\rtarr (M,M-x)$ factors through $(M-x,M-x)$ and therefore $s$ maps to zero in $H_n(M,M-x)$. This proves our claim. Now consider the following diagram, where $y\in U- U\cap V$: $$\diagram & & H_n(U\cup V) \rto \dlto & H_n(M) \dto^{0}\\ H_n(V,U\cap V)\dto_{\pa} \rto & H_n(U\cup V, U\cap V) \dlto^{\pa} \rrto & & H_n(M,M-y) \\ \tilde{H}_{n-1}(U\cap V) \dto_{i_*} & H_n(U,U\cap V) \lto^{\pa} \rrto \uto & & H_n(U,U-y) \uto_{\iso}\\ \tilde{H}_{n-1}(V). & & & \\ \enddiagram$$ Let $r\in \ker\,i_*$. Since $\tilde{H}_{n-1}(U)=0$, the bottom map $\pa$ is an epimorphism and there exists $s\in H_n(U,U\cap V)$ such that $\pa(s)=r$. We claim that $s$ maps to zero in $H_n(U,U-y)$ for every $y\in U-(U\cap V)$. By the previous lemma, this will imply that $s=0$ and thus $r=0$, so that $i_*$ is indeed a monomorphism. Since $i_*(r)=0$, there exists $t\in H_n(V,U\cap V)$ such that $\pa(t)=r$. Let $s'$ and $t'$ be the images of $s$ and $t$ in $H_n(U\cup V,U\cap V)$. Then $\pa(s'-t')=0$, hence there exists $w\in H_n(U\cup V)$ that maps to $s'-t'$. Since $w$ maps to zero in $H_n(M,M-y)$, so does $s'-t'$. Since the map $(V,U\cap V)\rtarr (M,M-y)$ factors through $(M-y,M-y)$, $t$ and thus also $t'$ maps to zero in $H_n(M,M-y)$. Therefore $s'$ maps to zero in $H_n(M,M-y)$ and thus $s$ maps to zero in $H_n(U,U-y)$, as claimed. \end{proof} \section{The proof of the Poincar\'e duality theorem} Let $M$ be an $R$-oriented $n$-manifold, not necessarily compact. Unless otherwise specified, we take homology and cohomology with coefficients in a given $R$-module $\pi$ in this section. Remember that homology is a covariant functor with compact supports. Cohomology is a contravariant functor, and it does not have compact supports. We would like to prove the Poincar\'e duality theorem by inductive comparisons of Mayer-Vietoris sequences, and the opposite variance of homology and cohomology makes it unclear how to proceed. To get around this, we introduce a variant of cohomology that does have compact supports and has enough covariant functoriality to allow us to proceed by comparisons of Mayer-Vietoris sequences. Consider the set $\sK$ of compact subspaces $K$ of $M$. This set is directed under inclusion; to conform with our earlier discussion of colimits, we may view $\sK$ as a category whose objects are the compact subspaces $K$ and whose maps are the inclusions between them. We define $$H^q_c(M) = \colim H^q(M,M-K),$$ where the colimit is taken with respect to the homomorphisms $$H^q(M,M-K) \rtarr H^q(M,M-L)$$ induced by the inclusions $(M,M-L)\subset (M,M-K)$ for $K\subset L$. This is the cohomology of $M$ with compact supports.\index{cohomology with compact supports}\index{compactly supported cohomology} Intuitively, thinking in terms of singular cohomology, its elements are represented by cocycles that vanish off some compact subspace. A map $f:M\rtarr N$ is said to be proper if $f^{-1}(L)$ is compact in $M$ when $L$ is compact in $N$. This holds, for example, if $f$ is the inclusion of a closed subspace. For such $f$, we obtain an induced homomorphism $f^*: H^*_c(N)\rtarr H^*_c(M)$ in an evident way. However, we shall make no use of this contravariant functoriality. What we shall use is a kind of covariant functoriality that will allow us to compare long exact sequences in homology and cohomology. Explicitly, for an open subspace $U$ of $M$, we obtain a homomorphism $H^q_c(U)\rtarr H^q_c(M)$ by passage to colimits from the excision isomorphisms $$ H^q(U,U-K) \rtarr H^q(M,M-K)$$ for compact subspaces $K$ of $U$. For each compact subspace $K$ of $M$, the $R$-orientation of $M$ determines a fundamental class $z_K\in H_n(M,M-K;R)$. Taking the relative cap product with $z_K$, we obtain a duality homomorphism $$D_K: H^p(M,M-K) \rtarr H_{n-p}(M).$$ If $K\subset L$, the following diagram commutes: $$\diagram H^p(M,M-K) \rrto \drto_{D_K} & & H^p(M,M-L) \dlto^{D_L}\\ & H_{n-p}(M). & \\ \enddiagram$$ We may therefore pass to colimits to obtain a duality homomorphism $$D: H^p_c(M) \rtarr H_{n-p}(M).$$ If $U$ is open in $M$ and is given the induced $R$-orientation, then the following naturality diagram commutes: $$\diagram H^p_c(U) \dto \rto^(0.43){D} & H_{n-p}(U) \dto \\ H^p_c(M) \rto_(0.4){D} & H_{n-p}(M).\\ \enddiagram$$ If $M$ itself is compact, then $M$ is cofinal among the compact subspaces of $M$. Therefore $H^p_c(M) = H^p(M)$, and the present duality map $D$ coincides with that of the Poincar\'e duality theorem as originally stated. We shall prove a generalization to not necessarily compact manifolds. \begin{thm}[Poincar\'e duality]\index{Poincare duality theorem@Poincar\'e duality theorem} Let $M$ be an $R$-oriented $n$-manifold. Then \linebreak $D: H^p_c(M)\rtarr H_{n-p}(M)$ is an isomorphism. \end{thm} \begin{proof} We shall prove that $D: H^p_c(U)\rtarr H_{n-p}(U)$ is an isomorphism for every open subspace $U$ of $M$. The proof proceeds in five steps. \begin{proof}[Step 1] {\em The result holds for any coordinate chart $U$.}\\ We may take $U=M=\bR^n$. The compact cubes $K$ are cofinal among the compact subspaces of $\bR^n$. For such $K$ and for $x\in K$, $$H^p(\bR^n,\bR^n-K)\iso H^p(\bR^n,\bR^n-x)\iso \tilde{H}^{p-1}(S^{n-1})\iso \tilde{H}^p(S^n).$$ The maps of the colimit system defining $H^p_c(\bR^n)$ are clearly isomorphisms. By the definition of the cap product, we see that $D: H^n(\bR^n,\bR^n-x)\rtarr H_0(\bR^n)$ is an isomorphism. Therefore $D_K$ is an isomorphism for every compact cube $K$ and so $D: H^n_c(\bR^n)\rtarr H_0(\bR^n)$ is an isomorphism. \end{proof} \begin{proof}[Step 2] {\em If the result holds for open subspaces $U$ and $V$ and their intersection, then it holds for their union.}\\ Let $W=U\cap V$ and $Z=U\cup V$. The compact subspaces of $Z$ that are unions of a compact subspace $K$ of $U$ and a compact subspace $L$ of $V$ are cofinal among all of the compact subspaces of $Z$. For such $K$ and $L$, we have the following commutative diagram with exact rows. We let $J=K\cap L$ and $N=K\cup L$, and we write $U_K = (U,U-K)$, and so on, to abbreviate notation. \begin{small} $$\diagram \rto & H^p(Z_J) \rto \dto_{\iso}& H^p(Z_K)\oplus H^p(Z_L) \rto \dto^{\iso} & H^p(Z_N) \rto \ddouble & H^{p+1}(Z_J) \rto \dto^{\iso} & \\ \rto & H^p(W_J) \rto \dto_D & H^p(U_K)\oplus H^p(V_L) \rto \dto^{D\oplus D} & H^p(Z_N) \rto \dto^D & H^{p+1}(W_J) \rto \dto^D & \\ \rto & H_{n-p}(W) \rto & H_{n-p}(U)\oplus H_{n-p}(V) \rto & H_{n-p}(Z) \rto & H_{n-p-1}(W)\rto &\\ \enddiagram$$ \end{small} The top row is the relative Mayer-Vietoris sequence of the triad $(Z;Z-K,Z-L)$. The middle row results from the top row by excision isomorphisms. The bottom row is the absolute Mayer-Vietoris sequence of the triad $(Z;U,V)$. The left two squares commute by naturality. The right square commutes by a diagram chase from the definition of the cap product. The entire diagram is natural with respect to pairs $(K,L)$. We obtain a commutative diagram with exact rows on passage to colimits, and the conclusion follows by the five lemma. \end{proof} \begin{proof}[Step 3] {\em If the result holds for each $U_i$ in a totally ordered set of open subspaces $\sset{U_i}$, then it holds for the union $U$ of the $U_i$.}\\ Any compact subspace $K$ of $U$ is contained in a finite union of the $U_i$ and therefore in one of the $U_i$. Since homology is compactly supported, it follows that $\colim H_{n-p}(U_i)\iso H_{n-p}(U)$. On the cohomology side, we have \begin{eqnarray*} \colim_i\,H^p_c(U_i) & = & \colim_i\colim_{\sset{K| K\subset U_i}} H^p(U_i,U_i-K) \\ & \iso & \colim_{\sset{K\subset U}}\colim_{\sset{i|K\subset U_i}} H^p(U_i,U_i-K) \\ & \iso & \colim_{\sset{K\subset U}} H^p(U,U-K) = H^p_c(U). \end{eqnarray*} Here the first isomorphism is an (algebraic) interchange of colimits isomorphism: both composite colimits are isomorphic to $\colim H^p_c(U_i,U_i-K)$, where the colimit runs over the pairs $(K,i)$ such that $K\subset U_i$. The second isomorphism holds since $\colim_{\sset{i|K\subset U_i}} H^p(U_i,U_i-K)\iso H^p(U,U-K)$ because the colimit is taken over a system of inverses of excision isomorphisms. The conclusion follows since a colimit of isomorphisms is an isomorphism. \end{proof} \begin{proof}[Step 4] {\em The result holds if $U$ is an open subset of a coordinate neighborhood.}\\ We may take $M=\bR^n$. If $U$ is a convex subset of $\bR^n$, then $U$ is homeomorphic to $\bR^n$ and Step 1 applies. Since the intersection of two convex sets is convex, it follows by induction from Step 2 that the conclusion holds for any finite union of convex open subsets of $\bR^n$. Any open subset $U$ of $\bR^n$ is the union of countably many convex open subsets. By ordering them and letting $U_i$ be the union of the first $i$, we see that the conclusion for $U$ follows from Step 3. \end{proof} \begin{proof}[Step 5] {\em The result holds for any open subset $U$ of $M$}.\\ We may as well take $M=U$. By Step 3, we may apply Zorn's lemma to conclude that there is a maximal open subset $V$ of $M$ for which the conclusion holds. If $V$ is not all of $M$, say $x\not\in V$, we may choose a coordinate chart $U$ such that $x\in U$. By Steps 2 and 4, the result holds for $U\cup V$, contradicting the maximality of $V$. \end{proof} This completes the proof of the Poincar\'e duality theorem. \end{proof} \section{The orientation cover} There is an orientation cover\index{orientation cover} of a manifold that helps illuminate the notion of orientability. For the moment, we relax the requirement that the total space of a cover be connected. Here we take homology with integer coefficients. \begin{prop} Let $M$ be a connected $n$-manifold. Then there is a $2$-fold cover $p:\tilde{M}\rtarr M$ such that $\tilde{M}$ is connected if and only if $M$ is not orientable.\index{orientable} \end{prop} \begin{proof} Define $\tilde{M}$ to be the set of pairs $(x,\al)$, where $x\in M$ and where $\al\in H_n(M,M-x)\iso \bZ$ is a generator. Define $p(x,\al)=x$. If $U\subset M$ is open and $\be\in H_n(M,M-U)$ is a fundamental class of $M$ at $U$, define $$\langle U,\be\rangle = \sset{(x,\al)|x\in U \tand \be \ \text{maps to}\ \al}.$$ The sets $\langle U,\be\rangle$ form a base for a topology on $\tilde{M}$. In fact, if $(x,\al)\in \langle U,\be\rangle \cap \langle V,\ga\rangle$, we can choose a coordinate neighborhood $W\subset U\cap V$ such that $x\in W$. There is a unique class $\al'\in H_n(M,M-W)$ that maps to $\al$, and both $\be$ and $\ga$ map to $\al'$. Therefore $$\langle W,\al'\rangle \subset \langle U,\be\rangle \cap \langle V,\ga\rangle.$$ Clearly $p$ maps $\langle U,\be\rangle$ homeomorphically onto $U$ and $$p^{-1}(U) = \langle U,\be\rangle \cup \langle U,-\be\rangle.$$ Therefore $\tilde{M}$ is an $n$-manifold and $p$ is a $2$-fold cover. Moreover, $\tilde{M}$ is oriented. Indeed, if $U$ is a coordinate chart and $(x,\al)\in \langle U,\be\rangle$, then the following maps all induce isomorphisms on passage to homology: $$\diagram (\tilde{M},\tilde{M}-\langle U,\be\rangle) \dto & (M,M-U) \dto \\ (\tilde{M},\tilde{M}-(x,\al)) & (M,M-x) \\ (\langle U,\be\rangle, \langle U,\be\rangle -(x,\al)) \uto \rto^(0.6)p_(0.6){\iso} & (U,U-x). \uto \\ \enddiagram$$ Via the diagram, $\be\in H_n(M,M-U)$ specifies an element $\tilde{\be}\in H_n(\tilde{M},\tilde{M}-\langle U,\be\rangle)$, and $\tilde{\be}$ is independent of the choice of $(x,\al)$. These classes are easily seen to specify an orientation of $\tilde{M}$. Essentially by definition, an orientation of $M$ is a cross section $s: M \rtarr \tilde{M}$: if $s(U) = \langle U,\be\rangle$, then these $\be$ specify an orientation. Given one section $s$, changing the signs of the $\be$ gives a second section $-s$ such that $\tilde{M}= \im(s)\amalg \im(-s)$, showing that $\tilde{M}$ is not connected if $M$ is oriented. \end{proof} The theory of covering spaces gives the following consequence. \begin{cor} If $M$ is simply connected, or if $\pi_1(M)$ contains no subgroup of index $2$, then $M$ is orientable. If $M$ is orientable, then $M$ admits exactly two orientations. \end{cor} \begin{proof} If $M$ is not orientable, then $p_*(\pi_1(\tilde{M}))$ is a subgroup of $\pi_1(M)$ of index $2$. This implies the first statement, and the second statement is clear. \end{proof} We can use homology with coefficients in a commutative ring $R$ to construct an analogous $R$-orientation cover.\index{Rorientation cover@$R$-orientation cover} It depends on the units of $R$. For example, if $R=\bZ_2$, then the $R$-orientation cover is the identity map of $M$ since there is a unique unit in $R$. This reproves the obvious fact that any manifold is $\bZ_2$-oriented. The evident ring homomorphism $\bZ\rtarr R$ induces a natural homomorphism $H_*(X;\bZ)\rtarr H_*(X;R)$, and we see immediately that an orientation of $M$ induces an $R$-orientation of $M$ for any $R$. \vspace{.1in} \begin{center} PROBLEMS \end{center} \begin{enumerate} \item Prove: there is no homotopy equivalence $f: \bC P^{2n}\rtarr \bC P^{2n}$ that reverses orientation (induces multiplication by $-1$ on $H_{4n}(\bC P^{2n})$). \end{enumerate} In the problems below, $M$ is assumed to be a compact connected $n$-manifold (without boundary), where $n\geq 2$. \begin{enumerate} \item[2.] Prove that if $M$ is a Lie group, then $M$ is orientable. \item[3.] Prove that if $M$ is orientable, then $H_{n-1}(M; \bZ)$ is a free Abelian group. \item[4.] Prove that if $M$ is not orientable, then the torsion subgroup of $H_{n-1}(M; \bZ)$ is cyclic of order $2$ and $H_n(M; \bZ_q)$ is zero if $q$ is odd and is cyclic of order $2$ if $q$ is even. (Hint: use universal coefficients and the transfer homomorphism of the orientation cover.) \item[5.] Let $M$ be oriented with fundamental class $z$. Let $f: S^n\rtarr M$ be a map such that $f_*(i_n) = qz$, where $i_n \in H_n(S^n; \bZ)$ is the fundamental class and $q \neq 0$. \begin{enumerate} \item[(a)] Show that $f_*: H_*(S^n; \bZ_p) \rtarr H_*(M; \bZ_p)$ is an isomorphism if $p$ is a prime that does not divide $q$. \item[(b)] Show that multiplication by $q$ annihilates $H_i(M; \bZ)$ if $1 \leq i \leq n-1$. \end{enumerate} \item[6.] \begin{enumerate} \item[(a)] Let $M$ be a compact $n$-manifold. Suppose that $M$ is homotopy equivalent to $\SI Y$ for some connected based space $Y$. Deduce that $M$ has the same integral homology groups as $S^n$. (Hint: use the vanishing of cup products on $\tilde H^*(\SI Y)$ and Poincar\'e duality, treating the cases $M$ orientable and $M$ non-orientable separately.) \item[(b)] Deduce that $M$ is homotopy equivalent to $S^n$. Does it follow that $Y$ is homotopy equivalent to $S^{n-1}$? \end{enumerate} \item[7.]* Essay: The singular cohomology $H^*(M;\bR)$ is isomorphic to the de Rham cohomology of $M$. Why is this plausible? Sketch proof? \end{enumerate} \clearpage \thispagestyle{empty} \chapter{The index of manifolds; manifolds with boundary} The Poincar\'e duality theorem imposes strong constraints on the Euler characteristic of a manifold. It also leads to new invariants, most notably the index. Moreover, there is a relative version of Poincar\'e duality in the context of manifolds with boundary, and this leads to necessary algebraic conditions on the cohomology of a manifold that must be satisfied if it is to be a boundary. In particular, the index of a compact oriented $4n$-manifold $M$ is zero if $M$ is a boundary. We shall later outline the theory of cobordism, which leads to necessary {\em and sufficient} algebraic conditions for a manifold to be a boundary. \section{The Euler characteristic of compact manifolds} The Euler characteristic\index{Euler characteristic!of a space} $\ch (X)$ of a space with finitely generated homology is defined by $$\ch (X) = \textstyle{\sum}_i (-1)^i \ \text{rank}\ H_i(X;\bZ).$$ The universal coefficient theorem implies that $$\ch (X) = \textstyle{\sum}_i (-1)^i \dim H_i(X;F)$$ for {\em any} field of coefficients $F$. Examination of the relevant short exact sequences shows that $$\ch (X) = \textstyle{\sum}_i (-1)^i \ \text{rank}\ C_i(X;\bZ)$$ for {\em any} decomposition of $X$ as a finite CW complex. The verifications of these statements are immediate from earlier exercises. Now consider a compact oriented $n$-manifold. Recall that we take it for granted that $M$ can be decomposed as a finite CW complex, so that each $H_i(M;\bZ)$ is finitely generated. By the universal coefficient theorem and Poincar\'e duality, we have $$H_i(M;F)\iso H^i(M;F)\iso H_{n-i}(M;F)$$ for any field $F$. We may take $F=\bZ_2$, and so dispense with the requirement that $M$ be oriented. If $n$ is odd, the summands of $\ch(M)$ cancel in pairs, and we obtain the following conclusion. \begin{prop} If $M$ is a compact manifold of odd dimension, then $\ch(M)=0$. \end{prop} If $n=2m$ and $M$ is oriented, then $$\ch(M) = \textstyle{\sum}_{i=0}^{m-1} (-1)^i 2 \dim H_i(M) + (-1)^m \dim H_m(M)$$ for any field $F$ of coefficients. Let us take $F=\bQ$. Of course, we can replace homology by cohomology in the definition and formulas for $\ch(M)$. The middle dimensional cohomology group $H^m(M)$ plays a particularly important role. Recall that we have the cup product pairing\index{cup product pairing} $$\ph: H^m(M)\ten H^m(M) \rtarr \bQ$$ specified by $\ph(\al,\be) = \langle\al\cup\be,z\rangle$. This pairing is nonsingular. Since $\al\cup \be =(-1)^m\be\cup\al$, it is skew symmetric if $m$ is odd and is symmetric if $m$ is even. When $m$ is odd, we obtain the following conclusion. \begin{prop} If $M$ is a compact oriented $n$-manifold, where $n\equiv 2\ \text{mod}\ 4$, then $\ch(M)$ is even. \end{prop} \begin{proof} It suffices to prove that $\dim H^{m}(M)$ is even, where $n=2m$, and this is immediate from the following algebraic observation. \end{proof} \begin{lem} Let $F$ be a field of characteristic $\neq 2$, $V$ be a finite dimensional vector space over $F$, and $\ph: V\times V \rtarr F$ be a nonsingular skew symmetric bilinear form. Then $V$ has a basis $\sset{x_1,\ldots\!,x_r, y_1,\ldots\!,y_r}$ such that $\ph(x_i,y_i)=1$ for $1\leq i\leq r$ and $\ph(z,w)=0$ for all other pairs of basis elements $(z,w)$. Therefore the dimension of $V$ is even. \end{lem} \begin{proof} We proceed by induction on $\dim V$, and we may assume that $V\neq 0$. Since $\ph(x,y)=-\ph(y,x)$, $\ph(x,x)=0$ for all $x\in V$. Choose $x_1\neq 0$. Certainly there exists $y_1$ such that $\ph(x_1,y_1)=1$, and $x_1$ and $y_1$ are then linearly independent. Define $$W=\sset{x|\ph(x,x_1) = 0 \tand \ph(x,y_1)=0}\subset V.$$ That is, $W$ is the kernel of the homomorphism $\ps: V\rtarr F\times F$ specified by $\ps(x)= (\ph(x,x_1),\ph(x,y_1))$. Since $\ps(x_1)=(0,1)$ and $\ps(y_1)=(-1,0)$, $\ps$ is an epimorphism. Thus $\dim W = \dim V -2$. Since $\ph$ restricts to a nonsingular skew symmetric bilinear form on $W$, the conclusion follows from the induction hypothesis. \end{proof} \section{The index of compact oriented manifolds} To study manifolds of dimension $4k$, we consider an analogue for symmetric bilinear forms of the previous algebraic lemma. Since we will need to take square roots, we will work over $\bR$. \begin{lem} Let $V$ be a finite dimensional real vector space and $\ph: V\times V \rtarr \bR$ be a nonsingular symmetric bilinear form. Define $q(x)=\ph(x,x)$. Then $V$ has a basis $\sset{x_1,\ldots\!,x_r, y_1,\ldots\!,y_s}$ such that $\ph(z,w)=0$ for all pairs $(z,w)$ of distinct basis elements, $q(x_i)=1$ for $1\leq i\leq r$ and $q(y_j)=-1$ for $1\leq j\leq s$. The number $r-s$ is an invariant of $\ph$, called the signature\index{signature} of $\ph$. \end{lem} \begin{proof} We proceed by induction on $\dim V$, and we may assume that $V\neq 0$. Clearly $q(rx)=r^2q(x)$. Since we can take square roots in $\bR$, we can choose $x_1\in V$ such that $q(x_1)=\pm 1$. Define $\ps: V\rtarr \bR$ by $\ps(x)=\ph(x,x_1)$ and let $W=\ker \ps$. Since $\ps(x_1)=\pm 1$, $\ps$ is an epimorphism and $\dim W=\dim V-1$. Since $\ph$ restricts to a nonsingular symmetric bilinear form on $W$, the existence of a basis as specified follows directly from the induction hypothesis. Invariance means that the integer $r-s$ is independent of the choice of basis on which $q$ takes values $\pm 1$, and we leave the verification to the reader. \end{proof} \begin{defn} Let $M$ be a compact oriented $n$-manifold. If $n=4k$, define the index\index{index} of $M$, denoted $I(M)$, to be the signature of the cup product form $H^{2k}(M;\bR)\ten H^{2k}(M;\bR)\rtarr \bR$. If $n\,\not\!\equiv\,0 \ \text{mod}\ 4$, define $I(M)=0$. \end{defn} The Euler characteristic and index are related by the following congruence. \begin{prop} For any compact oriented $n$-manifold, $\ch(M)\equiv I(M)\ \text{mod}\ 2$. \end{prop} \begin{proof} If $n$ is odd, then $\ch(M)=0$ and $I(M)=0$. If $n\equiv 2\ \text{mod}\ 4$, then $\ch(M)$ is even and $I(M)=0$. If $n=4k$, then $I(M) = r-s$, where $r+s = \dim H^{2k}(M;\bR) \equiv \ch(M)\ \text{mod}\ 2$. \end{proof} Observe that the index of $M$ changes sign if the orientation of $M$ is reversed. We write $-M$ for $M$ with the reversed orientation, and then $I(-M)=-I(M)$. We also have the following algebraic identities. Write $H^*(M)=H^*(M;\bR)$. \begin{lem} If $M$ and $M'$ are compact oriented $n$-manifolds, then $$I(M\amalg M')=I(M)+I(M'),$$ where $M\amalg M'$ is given the evident orientation induced from those of $M$ and $M'$. \end{lem} \begin{proof} There is nothing to prove unless $n=4k$, in which case $$H^{2k}(M\amalg M')=H^{2k}(M)\times H^{2k}(M').$$ Clearly the cup product of an element of $H^*(M)$ with an element of $H^*(M')$ is zero, and the cup product form on $H^{2k}(M\amalg M')$ is given by $$\ph((x,x'),(y,y')) = \ph(x,y)+\ph(x',y')$$ for $x,y\in H^{2k}(M)$ and $x',y'\in H^{2k}(M')$. The conclusion follows since the signature of a sum of forms is the sum of the signatures. \end{proof} \begin{lem} Let $M$ be a compact oriented $m$-manifold and $N$ be a compact oriented $n$-manifold. Then $$I(M\times N)=I(M)\cdot I(N),$$ where $M\times N$ is given the orientation induced from those of $M$ and $N$. \end{lem} \begin{proof} We must first make sense of the induced orientation on $M\times N$. For CW pairs $(X,A)$ and $(Y,B)$, we have an identification of CW complexes $$ (X\times Y)/(X\times B\cup A\times Y)\iso (X/A)\sma (Y/B)$$ and therefore an isomorphism $$ C_*(X\times Y,\, X\times B\cup A\times Y)\iso C_*(X,A)\ten C_*(Y,B).$$ This implies a relative K\"{u}nneth theorem\index{Kunneth theorem@K\"unneth theorem!relative} for arbitrary pairs $(X,A)$ and $(Y,B)$. For subspaces $K\subset M$ and $L\subset N$, $$(M\times N, M\times N-K\times L) = (M\times N, M\times (N-L) \cup (M-K)\times N).$$ In particular, for points $x\in M$ and $y\in Y$, $$(M\times N, M\times N-(x,y)) = (M\times N, M\times (N-y) \cup (M-x)\times N).$$ Therefore fundamental classes $z_K$ of $M$ at $K$ and $z_L$ of $N$ at $L$ determine a fundamental class $z_{K\times L}$ of $M\times N$ at $K\times L$. In particular, the image under $H_m(M)\ten H_n(N)\rtarr H_{m+n}(M\times N)$ of the tensor product of fundamental classes of $M$ and $N$ is a fundamental class of $M\times N$. Turning to the claimed product formula, we see that there is nothing to prove unless $m+n=4k$, in which case $$H^{2k}(M\times N)=\sum_{i+j=2k} H^i(M)\ten H^j(N).$$ The cup product form is given by $$\ph(x\ten y, x'\ten y') = (-1)^{(\deg y)(\deg x')+ mn}\langle x\cup x',z_M\rangle \langle y\cup y',z_N\rangle$$ for $x, x'\in H^*(M)$ and $y,y'\in H^*(N)$. If $m$ and $n$ are odd, then the signature of this form is zero. If $m$ and $n$ are even, then this form is the sum of the tensor product of the cup product forms on the middle dimensional cohomology groups of $M$ and $N$ and a form whose signature is zero. Here, if $m$ and $n$ are congruent to 2 mod 4, the signature is zero since the lemma of the previous section implies that the signature of the tensor product of two skew symmetric forms is zero. When $m$ and $n$ are congruent to 0 mod 4, the conclusion holds since the signature of the tensor product of two symmetric forms is the product of their signatures. We leave the detailed verifications of these algebraic statements as exercises for the reader. \end{proof} \section{Manifolds with boundary} Let $\bH^n=\sset{(x_1,\ldots\!,x_n)|x_n\geq 0}$ be the upper half-plane in $\bR^n$. Recall that an $n$-manifold with boundary\index{manifold with boundary} is a Hausdorff space $M$ having a countable basis of open sets such that every point of $M$ has a neighborhood homeomorphic to an open subset of $\bH^n$. A point $x$ is an interior point if it has a neighborhood homeomorphic to an open subset of $\bH^n-\pa \bH^n\iso \bR^n$; otherwise it is a boundary point. It is a fact called ``invariance of domain''\index{invariance of domain} that if $U$ and $V$ are homeomorphic subspaces of $\bR^n$ and $U$ is open, then $V$ is open. Therefore, a homeomorphism of an open subspace of $\bH^n$ onto an open subspace of $\bH^n$ carries boundary points to boundary points. We denote the boundary\index{boundary of a manifold} of an $n$-manifold $M$ by $\pa M$. Thus $M$ is a manifold without boundary if $\pa M$ is empty; $M$ is said to be closed\index{closed manifold} if, in addition, it is compact. The space $\pa M$ is an $(n-1)$-manifold without boundary. It is a fundamental question in topology to determine which closed manifolds are boundaries. The question makes sense with varying kinds of extra structure. For example, we can ask whether or not a smooth (= differentiable) closed manifold is the boundary of a smooth manifold (with the induced smooth structure). Numerical invariants in algebraic topology give criteria. One such criterion is given by the following consequence of the Poincar\'e duality theorem. Remember that $\ch(M)=0$ if $M$ is a closed manifold of odd dimension. \begin{prop} If $M=\pa W$, where $W$ is a compact $(2m+1)$-manifold, then $\ch(M)=2\ch (W)$. \end{prop} \begin{proof} The product $W\times I$ is a $(2m+2)$-manifold with $$\pa(W\times I) = (W\times \sset{0}) \cup (M\times I) \cup (W\times \sset{1}).$$ Let $U=\pa(W\times I)-(W\times \sset{1})$ and $V=\pa(W\times I)-(W\times \sset{0})$. Then $U$ and $V$ are open subsets of $\pa(W\times I)$. Clearly $U$ and $V$ are both homotopy equivalent to $W$ and $U\cap V$ is homotopy equivalent to $M$. We have the Mayer-Vietoris sequence $$\diagram H_{i+1}(U\cup V) \rto \ddouble & H_i(U\cap V)\rto \dto^{\iso} & H_i(U)\oplus H_i(V) \rto \dto^{\iso} & H_i(U\cup V) \ddouble\\ H_{i+1}(\pa(W\times I)) \rto & H_i(M)\rto & H_i(W)\oplus H_i(W) \rto & H_i(\pa(W\times I)).\\ \enddiagram$$ Therefore $2\ch(W)=\ch(M)+\ch(\pa(W\times I))$. However, $\ch(\pa(W\times I))=0$ since $\pa(W\times I)$ is a closed manifold of odd dimension. \end{proof} \begin{cor} If $M=\pa W$ for a compact manifold $W$, then $\ch(M)$ is even. \end{cor} For example, since $\ch(\bR P^{2m})=1$ and $\ch(\bC P^n)=n+1$, this criterion shows that $\bR P^{2m}$ and $\bC P^{2m}$ cannot be boundaries. Notice that we have proved that these are not boundaries of topological manifolds, let alone of smooth ones. \section{Poincar\'e duality for manifolds with boundary} The index gives a more striking criterion: if a closed oriented $4k$-manifold $M$ is the boundary of a (topological) manifold, then $I(M)=0$. To prove this, we must first obtain a relative form of the Poincar\'e duality theorem applicable to manifolds with boundary. We let $M$ be an $n$-manifold with boundary, $n>0$, throughout this section, and we let $R$ be a given commutative ring. We say that $M$ is $R$-orientable\index{Rorientable@$R$-orientable} (or orientable\index{orientable} if $R=\bZ$) if its interior $\cir{M}=M-\pa M$ is $R$-orientable; similarly, an $R$-orientation\index{Rorientation@$R$-orientation} of $M$ is an $R$-orientation of its interior. To study these notions, we shall need the following result, which is intuitively clear but is somewhat technical to prove. In the case of smooth manifolds, it can be seen in terms of inward-pointing unit vectors of the normal line bundle of the embedding $\pa M\rtarr M$. \begin{thm}[Topological collaring]\index{topological collar} There is an open neighborhood $V$ of $\pa M$ in $M$ such that the identification $\pa M = \pa M\times \sset{0}$ extends to a homeomorphism $V\iso \pa M\times [0,1)$. \end{thm} It follows that the inclusion $\cir{M}\rtarr M$ is a homotopy equivalence and the inclusion $\pa M\rtarr M$ is a cofibration. We take homology with coefficients in $R$ in the next two results. \begin{prop} An $R$-orientation of $M$ determines an $R$-orientation of $\pa M$. \end{prop} \begin{proof} Consider a coordinate chart $U$ of a point $x\in \pa M$. If $\dim M= n$, then $U$ is homeomorphic to an open half-disk in $\bH^n$. Let $V=\pa U = U\cap \pa M$ and let $y\in \cir{U}=U-V$. We have the following chain of isomorphisms: \begin{eqnarray*} H_n(\cir{M},\cir{M}-\cir{U}) & \iso & H_n(\cir{M},\cir{M}-y) \\ & \iso & H_n(M,M-y) \\ & \iso & H_n(M,M-\cir{U})\\ & \overto{\pa} & H_{n-1}(M-\cir{U},M-U) \\ & \iso & H_{n-1}(M-\cir{U},(M-\cir{U})-x)\\ & \iso & H_{n-1}(\pa M,\pa M-x)\\ & \iso & H_{n-1}(\pa M,\pa M-V). \end{eqnarray*} The first and last isomorphisms are restrictions of the sort that enter into the definition of an $R$-orientation, and the third isomorphism is similar. We see by use of a small boundary collar that the inclusion $(\cir{M},\cir{M}-y) \rtarr (M,M-y)$ is a homotopy equivalence, and that gives the second isomorphism. The connecting homomorphism is that of the triple $(M,M-\cir{U},M-U)$ and is an isomorphism since $H_*(M,M-U)\iso H_*(M,M)=0$. The isomorphism that follows comes from the observation that the inclusion $(M-\cir{U})-x \rtarr M-U$ is a homotopy equivalence, and the next to last isomorphism is given by excision of $\cir{M}-\cir{U}$. The conclusion is an easy consequence of these isomorphisms. \end{proof} \begin{prop} If $M$ is compact and $R$-oriented and $z_{\pa M}\in H_{n-1}(\pa M)$ is the fundamental class determined by the induced $R$-orientation on $\pa M$, then there is a unique element $z\in H_n(M,\pa M)$ such that $\pa z = z_{\pa M}$; $z$ is called the $R$-fundamental class\index{Rfundamental class@$R$-fundamental class} determined by the $R$-orientation of $M$. \end{prop} \begin{proof} Since $\cir{M}$ is a non-compact manifold without boundary and $\cir{M}\rtarr M$ is a homotopy equivalence, $H_n(M)\iso H_n(\cir{M})=0$ by the vanishing theorem. Therefore $\pa: H_n(M,\pa M)\rtarr H_{n-1}(\pa M)$ is a monomorphism. Let $V$ be a boundary collar and let $N=M-V$. Then $N$ is a closed subspace and a deformation retract of the $R$-oriented open manifold $\cir{M}$, and we have $$H_n(\cir{M},\cir{M}-N)\iso H_n(M,M-\cir{M}) = H_n(M,\pa M).$$ Since $M$ is compact, $N$ is a compact subspace of $\cir{M}$. Therefore the $R$-orientation of $\cir{M}$ determines a fundamental class in $H_n(\cir{M},\cir{M}-N)$. Let $z$ be its image in $H_n(M,\pa M)$. Then $z$ restricts to a generator of $H_n(M,M-y)\iso H_n(\cir{M},\cir{M}-y)$ for every $y\in \cir{M}$. Via naturality diagrams and the chain of isomorphisms in the previous proof, we see that $\pa z$ restricts to a generator of $H_{n-1}(\pa M,\pa M-x)$ for all $x\in \pa M$ and is the fundamental class determined by the $R$-orientation of $\pa M$. \end{proof} \begin{thm}[Relative Poincar\'e duality]\index{Poincare duality theorem@Poincar\'e duality theorem!relative} Let $M$ be a compact $R$-oriented $n$-\linebreak manifold with $R$-fundamental class $z\in H_n(M,\pa M;R)$. Then, with coefficients taken in any $R$-module $\pi$, capping with $z$ specifies duality isomorphisms $$ D: H^p(M,\pa M)\rtarr H_{n-p}(M) \ \ \tand \ \ D:H^p(M)\rtarr H_{n-p}(M,\pa M).$$ \end{thm} \begin{proof} The following diagram commutes by inspection of definitions: $$\diagram H^{p-1}(\pa M) \rto \dto_D & H^p(M,\pa M) \rto \dto^D & H^p(M) \rto \dto^D & H^p(\pa M) \dto^D\\ H_{n-p}(\pa M) \rto & H_{n-p}(M) \rto & H_{n-p}(M,\pa M) \rto & H_{n-p-1}(\pa M).\\ \enddiagram$$ Here $D$ for $\pa M$ is obtained by capping with $\pa z$ and is an isomorphism. By the five lemma, it suffices to prove that $D: H^p(M)\rtarr H_{n-p}(M,\pa M)$ is an isomorphism. To this end, let $N = M\cup_{\pa M} M$ be the ``double''\index{double of a manifold} of $M$ and let $M_1$ and $M_2$ be the two copies of $M$ in $N$. Clearly $N$ is a compact manifold without boundary, and it is easy to see that $N$ inherits an $R$-orientation from the orientation on $M_1$ and the negative of the orientation on $M_2$. Of course, $\pa M = M_1\cap M_2$. If $U$ is the union of $M_1$ and a boundary collar in $M_2$ and $V$ is the union of $M_2$ and a boundary collar in $M_1$, then we have a Mayer-Vietoris sequence for the triad $(N;U,V)$. Using the evident equivalences of $U$ with $M_1$, $V$ with $M_2$, and $U\cap V$ with $\pa M$, this gives the exact sequence in the top row of the following commutative diagram. The bottom row is the exact sequence of the pair $(N,\pa M)$, and the isomorphism results from the homeomorphism $N/\pa M\iso (M_1/\pa M) \vee (M_2/\pa M)$; we abbreviate $N_1=(M_1,\pa M)$ and $N_2=(M_2,\pa M)$: $$\diagram H^p(N) \rto \dto_D & H^p(M_1) \oplus H^p(M_2) \rto^{\ps} \dto^{D\oplus D} & H^p(\pa M) \dto^D \rto^{\DE} & H^{p+1}(N) \dto^D \\ H_{n-p}(N) \rto \ddouble & H_{n-p}(N_1)\oplus H_{n-p}(N_2) \rto \dto^{\iso} & H_{n-p-1}(\pa M) \ddouble \rto & H_{n-p-1}(N) \ddouble \\ H_{n-p}(N) \rto & H_{n-p}(N,\pa M) \rto & H_{n-p-1}(\pa M) \rto & H_{n-p-1}(N). \\ \enddiagram$$ The top left square commutes by naturality. In the top middle square, we have $\ps(x,y)=i_1^*(x)-i_2^*(y)$, where $i_1: \pa M\rtarr M_1$ and $i_2: \pa M\rtarr M_2$ are the inclusions. Since $D$ for $M_2$ is the negative of $D$ for $M_1$ under the identifications with $M$, the commutativity of this square follows from the relation $D\com i^* = \pa\com D: H^p(M)\rtarr H_{n-p-1}(\pa M)$, $i: \pa M\rtarr M$, which holds by inspection of definitions. For the top right square, $\DE$ is the the top composite in the diagram $$\diagram H^p(\pa M)\rto^(0.3){\de} \dto_D & H^{p+1}(M_1,\pa M)\iso H^{p+1}(N,M_2)\dto^D \rto & H^{p+1}(N)\dto^D \\ H_{n-p-1}(\pa M) \rto_{{i_1}_*} & H_{n-p-1}(M_1) \rto & H_{n-p-1}(N). \enddiagram$$ The right square commutes by naturality, and $D\com \de = {i_1}_*\com D$ by inspection of definitions. By the five lemma, since the duality maps $D$ for $N$ and $\pa M$ are isomorphisms, both maps $D$ between direct summands must be isomorphisms. The conclusion follows. \end{proof} \section{The index of manifolds that are boundaries} We shall prove the following theorem. \begin{thm} If $M$ is the boundary of a compact oriented $(4k+1)$-manifold, then $I(M)=0$. \end{thm} We first give an algebraic criterion for the vanishing of the signature of a form and then show that the cup product form on the middle dimensional cohomology of $M$ satisfies the criterion. \begin{lem} Let $W$ be a $n$-dimensional subspace of a $2n$-dimensional real vector space $V$. Let $\ph: V\times V\rtarr \bR$ be a nonsingular symmetric bilinear form such that $\ph: W\times W\rtarr \bR$ is identically zero. Then the signature of $\ph$ is zero. \end{lem} \begin{proof} Let $r$ and $s$ be as in the definition of the signature. Then $r+s=2n$ and we must show that $r=s$. We prove that $r\geq n$. Applied to the form $-\ph$, this will also give that $s\geq n$, implying the conclusion. We proceed by induction on $n$. Let $\sset{x_1,\ldots\!,x_n,z_1,\ldots\!,z_n}$ be a basis for $V$, where $\sset{x_1,\ldots\!,x_n}$ is a basis for $W$. Define $\tha: V\rtarr \bR^n$ and $\ps: V\rtarr \bR^n$ by $$\tha(x) = (\ph(x,x_1),\ldots\!,\ph(x,x_n)) \ \tand \ \ps(x) = (\ph(x,z_1),\ldots\!,\ph(x,z_n)).$$ Since $\ph$ is nonsingular, $\ker{\tha}\cap\ker{\ps}=0$. Since $\ker{\tha}$ and $\ker{\ps}$ each have dimension at least $n$, neither can have dimension more than $n$ and $\tha$ and $\ps$ must both be epimorphisms. Choose $y_1$ such that $\tha(y_1)=(1,0,\ldots\!,0)$. Let $q(x)=\ph(x,x)$ and note that $q(x)=0$ if $x\in W$. Since $q(x_1)=0$ and $\ph(x_1,y_1)=1$, $q(ax_1+y_1) = 2a+q(y_1)$ for $a\in\bR$. Taking $a=(1-q(y_1))/2$, we find $q(ax_1+y_1)=1$. If $n=1$, this gives $r\geq 1$ and completes the proof. If $n>1$, define $\om: V\rtarr \bR^2$ by $\om(x)=(\ph(x,x_1),\ph(x,y_1))$. Since $\om(x_1)=(0,1)$ and $\om(y_1)=(1,q(y_1))$, $\om$ is an epimorphism. Let $V'=\ker\om$ and let $W'\subset V'$ be the span of $\sset{x_2,\ldots\!,x_n}$. The restriction of $\ph$ to $V'$ satisfies the hypothesis of the lemma, and the induction hypothesis together with the construction just given imply that $r\geq n$. \end{proof} Take homology and cohomology with coefficients in $\bR$. \begin{lem} Let $M=\pa W$, where $W$ is a compact oriented $(4k+1)$-manifold, and let $i: M\rtarr W$ be the inclusion. Let $\ph: H^{2k}(M)\ten H^{2k}(M)\rtarr \bR$ be the cup product form. Then the image of $i^*: H^{2k}(W)\rtarr H^{2k}(M)$ is a subspace of half the dimension of $H^{2k}(M)$ on which $\ph$ is identically zero. \end{lem} \begin{proof} Let $z\in H_{4k+1}(W,M)$ be the fundamental class. For $\al,\be\in H^{2k}(W)$, $$\ph(i^*(\al),i^*(\be))=\langle i^*(\al\cup\be),\pa z\rangle =\langle \al\cup\be,i_*\pa z\rangle =0$$ since $i_*\pa=0$ by the long exact sequence of the pair $(W,M)$. Thus $\ph$ is identically zero on $\im i^*$. The commutative diagram with exact rows $$\diagram H^{2k}(W) \rto^{i^*} \dto_D & H^{2k}(M) \rto^{\de} \dto^D & H^{2k+1}(W,M) \dto^D\\ H_{2k+1}(W,M) \rto_{\pa} & H_{2k}(M) \rto_{i_*} & H_{2k}(W) \enddiagram$$ implies that $H^{2k}(M)\iso \im i^*\oplus \im\de\iso \im i^*\oplus \im i_*$. Since $i^*$ and $i_*$ are dual homomorphisms, $\im i^*$ and $\im i_*$ are dual vector spaces and thus have the same dimension. \end{proof} \vspace{.1in} \begin{center} PROBLEMS \end{center} Let $M$ be a compact connected $n$-manifold with boundary $\pa M$, where $n\geq 2$. \begin{enumerate} \item Prove: $\pa M$ is not a retract of $M$. \item Prove: if $M$ is contractible, then $\pa M$ has the homology of a sphere. \item Assume that $M$ is orientable. Let $n = 2m+1$ and let $K$ be the kernel of the homomorphism $H_m(\pa M) \rtarr H_m(M)$ induced by the inclusion, where homology is taken with coefficients in a field. Prove: $\dimÊÊ\,H_m(\pa M)Ê=Ê2\dimÊÊ\,K$. \end{enumerate} Let $n = 3$ in the rest of the problems. \begin{enumerate} \item[4.] Prove: if $M$ is orientable, $\pa M$ is empty, and $H_1(M; \bZ) = 0$, then $M$ has the same homology groups as a $3$-sphere. \item[5.] Prove: if $M$ is nonorientable and $\pa M$ is empty, then $H_1(M;\bZ)$ is infinite. \end{enumerate} (Hint for the last three problems: use the standard classification of closed $2$-manifolds and think about first homology groups.) \begin{enumerate} \item[6.] Prove: if $M$ is orientable and $H_1(M;\bZ) = 0$, then $\pa M$ is a disjoint union of $2$-spheres. \item[7.] Prove: if $M$ is orientable, $\pa M \neq\ph$, and $\pa M$ contains no $2$-spheres, then $H_1(M;\bZ)$ is infinite. \item[8.] Prove: if $M$ is nonorientable and $\pa M$ contains no $2$-spheres and no projective planes, then $H_1(M;\bZ)$ is infinite. \end{enumerate} \chapter{Homology, cohomology, and $K(\pi,n)$s} We have given an axiomatic definition of ordinary homology and cohomology, and we have shown how to realize the axioms by means of either cellular or singular chain and cochain complexes. We here give a homotopical way of constructing ordinary theories that makes no use of chains, whether cellular or singular. We also show how to construct cup and cap products homotopically. This representation of homology and cohomology in terms of Eilenberg-Mac\,Lane spaces is the starting point of the modern approach to homology and cohomology theory, and we shall indicate how theories that do not satisfy the dimension axiom can be represented. We shall also describe Postnikov systems, which give a way to approximate general (simple) spaces by weakly equivalent spaces built up out of Eilenberg-Mac\,Lane spaces. This is conceptually dual to the way that CW complexes allow the approximation of spaces by weakly equivalent spaces built up out of spheres. Finally, we present the important notion of cohomology operations and relate them to the cohomology of Eilenberg-Mac\,Lane spaces. \section{$K(\pi,n)$s and homology} Recall that a reduced homology theory on based CW complexes is a sequence of functors $\tilde{E}_q$ from the homotopy category of based CW complexes to the category of Abelian groups. Each $\tilde{E}_q$ must satisfy the exactness and additivity axioms, and there must be a natural suspension isomorphism. Up to isomorphism, ordinary reduced homology with coefficients in $\pi$ is characterized as the unique such theory that satisfies the dimension axiom: $\tilde{E}_0(S^0) = \pi$ and $\tilde{E}_q(S^0) = 0$ if $q\neq 0$. We proceed to construct such a theory homotopically. For based spaces $X$ and $Y$, we let $[X,Y]$ denote the set of based homotopy classes of based maps $X\rtarr Y$. Recall that we require Eilenberg-Mac\,Lane spaces $K(\pi,n)$ to have the homotopy\index{Eilenberg-Mac\,Lane space} types of CW complexes and that, up to homotopy equivalence, there is a unique such space for each $n$ and $\pi$. By a result of Milnor, if $X$ has the homotopy type of a CW complex, then so does $\OM X$. By the Whitehead theorem, we therefore have a homotopy equivalence $$\tilde{\si}: K(\pi,n)\rtarr \OM K(\pi,n+1).$$ This map is the adjoint of a map $$\si: \SI K(\pi,n) \rtarr K(\pi,n+1).$$ We may take the smash product of the map $\si$ with a based CW complex $X$ and use the suspension homomorphism on homotopy groups to obtain maps \begin{eqnarray*} \pi_{q+n}(X\sma K(\pi,n)) & \overto{\SI} & \pi_{q+n+1}(\SI(X\sma K(\pi,n))) \\ & = & \pi_{q+n+1}(X\sma \SI K(\pi,n)) \overto{(\id\sma\si)_*} \pi_{q+n+1}(X\sma K(\pi,n+1)). \end{eqnarray*} \begin{thm} For CW complexes $X$, Abelian groups $\pi$ and integers $n\geq 0$, there are natural isomorphisms\index{homology theory!ordinary} $$ \tilde{H}_q(X;\pi)\iso \colim_{n}\pi_{q+n}(X\sma K(\pi,n)).$$ \end{thm} It suffices to verify the axioms, and the dimension axiom is clear. If $X=S^0$, then $X\sma K(\pi,n)=K(\pi,n)$. Here the homotopy groups in the colimit system are zero if $q\neq 0$, and, if $q=0$, the colimit runs over a sequence of isomorphisms between copies of $\pi$. The verifications of the rest of the axioms are exercises in the use of the homotopy excision and Freudenthal suspension theorems, and it is worthwhile to carry out these exercises in greater generality. \begin{defn} A prespectrum\index{prespectrum} is a sequence of based spaces $T_n$, $n\geq 0$, and based maps $\si: \SI T_n \rtarr T_{n+1}$. \end{defn} The example at hand is the Eilenberg-Mac\,Lane prespectrum $\sset{K(\pi,n)}$. Another example is the ``suspension prespectrum''\index{suspension prespectrum} $\sset{\SI^n X}$ of a based space $X$; the required maps $\SI(\SI^nX) \rtarr \SI^{n+1}X$ are the evident identifications. When $X=S^0$, this is called the sphere prespectrum.\index{sphere prespectrum} \begin{thm} Let $\sset{T_n}$ be a prespectrum such that $T_n$ is $(n-1)$-connected and of the homotopy type of a CW complex for each $n$. Define\index{homology theory!reduced} $$\tilde{E}_q(X) = \colim_{n}\pi_{q+n}(X\sma T_n),$$ where the colimit is taken over the maps $$ \pi_{q+n}(X\sma T_n) \overto{\SI} \pi_{q+n+1}(\SI(X\sma T_n)) \iso \pi_{q+n+1}(X\sma \SI T_n) \overto{\id\sma\si} \pi_{q+n+1}(X\sma T_{n+1}). $$ Then the functors $\tilde{E}_q$ define a reduced homology theory on based CW complexes. \end{thm} \begin{proof} Certainly the $\tilde{E}$ are well defined functors from the homotopy category of based CW complexes to the category of Abelian groups. We must verify the exactness, additivity, and suspension axioms. Without loss of generality, we may take the $T_n$ to be CW complexes with one vertex and no other cells of dimension less than $n$. Then $X\sma T_n$ is a quotient complex of $X\times T_n$, and it too has one vertex and no other cells of dimension less than $n$. In particular, it is $(n-1)$-connected. If $A$ is a subcomplex of $X$, then the homotopy excision theorem implies that the quotient map $$(X\sma T_n,A\sma T_n) \rtarr ((X\sma T_n)/(A\sma T_n),*)\iso ((X/A)\sma T_n,*)$$ is a $(2n-1)$-equivalence. We may restrict to terms with $n>q-1$ in calculating $\tilde{E}_q(X)$, and, for such $q$, the long exact sequence of homotopy groups of the pair $(X\sma T_n,A\sma T_n)$ gives that the sequence $$\pi_{q+n}(A\sma T_n) \rtarr \pi_{q+n}(X\sma T_n) \rtarr \pi_{q+n}((X/A)\sma T_n)$$ is exact. Since passage to colimits preserves exact sequences, this proves the exactness axiom. We need some preliminaries to prove the additivity axiom. \begin{defn} Define the weak product\index{weak product} $\prod^{w}_{\, _i} Y_i$ of a set of based spaces $Y_i$ to be the subspace of $\prod_i Y_i$ consisting of those points all but finitely many of whose coordinates are basepoints. \end{defn} \begin{lem} For a set of based spaces $\sset{Y_i}$, the canonical map $$\textstyle{\sum}_i \pi_q(Y_i) \rtarr \pi_q(\textstyle{\prod}^w_{\, i} Y_i)$$ is an isomorphism. \end{lem} \begin{proof} The homotopy groups of $\prod^w_{\, i} Y_i$ are the colimits of the homotopy groups of the finite subproducts of the $Y_i$, and the conclusion follows. \end{proof} \begin{lem} If $\sset{Y_i}$ is a set of based CW complexes, then $\prod^w_{\, i}Y_i$ is a CW complex whose cells are the cells of the finite subproducts of the $Y_i$. If each $Y_i$ has a single vertex and no $q$-cells for $q n$. The system can be displayed diagrammatically as follows: $$\diagram & \vdots \dto & \\ & X_{n+1} \dto^{p_{n+1}} \rto^(0.29){k^{n+3}} & K(\pi_{n+2}(X),n+3) \\ X \urto^{\al_{n+1}} \ddrto^{\al_1} \rto^{\al_n} & X_n \rto^(0.27){k^{n+2}} \dto & K(\pi_{n+1}(X),n+2) \\ & \vdots \dto & \\ & X_1 \rto^(0.33){k^3} & K(\pi_{2}(X),3). & \\ \enddiagram$$ Our requirement that Eilenberg-Mac\,Lane spaces have the homotopy types of CW complexes implies (by a result of Milnor) that each $X_{n}$ has the homotopy type of a CW complex. The maps $\alpha _{n}$ induce a weak equivalence $X \rightarrow \lim X_{n}$, but the inverse limit generally will not have the homotopy type of a CW complex. The ``$k$-invariants'' $\sset{k^{n+2}}$ \index{kinvariants@$k$-invariants} that specify the system are to be regarded as cohomology classes \[ k^{n+2}\in H^{n+2}(X_{n};{\pi}_{n+1}(X)). \] These classes together with the homotopy groups $\pi_{n}(X)$ specify the weak homotopy type of $X$. We outline the proof of the following theorem. \begin{thm} A simple space $X$ of the homotopy type of a CW complex has a Postnikov system. \end{thm} \begin{proof} Assume inductively that $\alpha _{n}: X \rightarrow X_{n}$ has been constructed. A consequence of the homotopy excision theorem shows that the cofiber $C(\alpha _{n})$ is $(n+1)$-connected and satisfies \[ {\pi} _{n+2}(C(\alpha_{n}))={\pi} _{n+1}(X). \] More precisely, the canonical map $\et: F(\alpha _{n}) \rightarrow \Omega C(\alpha _{n})$ induces an isomorphism on ${\pi} _{q}$ for $q\leq n+1$. We construct \[ j: C(\alpha _{n}) \rightarrow K({\pi} _{n+1}(X),n+2) \] by inductively attaching cells to $C(\alpha _{n})$ to kill its higher homotopy groups. We take the composite of $j$ and the inclusion $X_{n} \subset C(\alpha _{n})$ to be the $k$-invariant $$k^{n+2}: X_n \rtarr K(\pi_{n+1}(X),n+2).$$ By our definition of a Postnikov system, we must define $X_{n+1}$ to be the homotopy fiber of $k^{n+2}$. Thus its points are pairs $(\omega ,x)$ consisting of a path $\omega : I\rightarrow K({\pi}_{n+1}(X),n+2)$ and a point $x\in X_{n}$ such that $\omega (0)=*$ and $\omega (1)=k^{n+2}(x)$. The map $p_{n+1}: X_{n+1} \rightarrow X_{n}$ is given by $p_{n+1}(\omega ,x)=x$, and the map $\alpha _{n+1}: X \rightarrow X_{n+1}$ is given by $\alpha _{n+1}(x)=(\omega (x),\alpha_{n}(x))$, where $\omega (x)(t) = j(x,1-t)$, $(x,1-t)$ being a point on the cone $CX \subset C(\alpha _{n})$. Clearly $p_{n+1}\com\alpha _{n+1} = \alpha _{n}$. It is evident that $\alpha _{n+1}$ induces an isomorphism on ${\pi}_{q}$ for $q\leq n$, and a diagram chase shows that this also holds for $q=n+1$. \end{proof} \section{Cohomology operations} Consider a ``represented functor''\index{represented functor} $k(X)=[X,Z]$ and another contravariant functor $k'$ from the homotopy category of based CW complexes to the category of sets. The following simple observation actually applies to represented functors on arbitrary categories. We shall use it to describe cohomology operations, but it also applies to describe many other invariants in algebraic topology, such as the characteristic classes of vector bundles. \begin{lem}[Yoneda]\index{Yoneda lemma} There is a canonical bijection between natural transformations $\PH: k\rtarr k'$ and elements $\ph\in k'(Z)$. \end{lem} \begin{proof} Given $\PH$, we define $\ph$ to be $\PH(\id)$, where $\id\in k(Z)=[Z,Z]$ is the identity map. Given $\ph$, we define $\PH: k(X)\rtarr k'(X)$ by the formula $\PH(f)=f^*(\ph)$. Here $f$ is a map $X\rtarr Z$, and it induces $f^*=k'(f):k'(Z)\rtarr k'(X)$. It is simple to check that these are inverse bijections. \end{proof} We are interested in the case when $k'$ is also represented, say $k'(X)=[X,Z']$. \begin{cor} There is a canonical bijection between natural transformations $\PH: [-,Z]\rtarr [-,Z']$ and elements $\ph\in [Z,Z']$. \end{cor} \begin{defn} Suppose given cohomology theories $\tilde{E}^*$ and $\tilde{F}^*$. A cohomology operation\index{cohomology operation} of type $q$ and degree $n$ is a natural transformation $\tilde{E}^q\rtarr \tilde{F}^{q+n}$. A stable cohomology operation\index{cohomology operation!stable} of degree $n$ is a sequence $\sset{\PH^q}$ of cohomology operations of type $q$ and degree $n$ such that the following diagram commutes for each $q$ and each based space $X$: $$\diagram \tilde{E}^q(X)\rto^{\PH^q} \dto_{\SI} & \tilde{E}^{q+n}(X) \dto^{\SI} \\ \tilde{E}^{q+1}(\SI X) \rto_(0.45){\PH^{q+1}} & \tilde{E}^{q+1+n}(\SI X). \\ \enddiagram$$ We generally abbreviate notation by setting $\PH^q=\PH$. \end{defn} In general, cohomology operations are only natural transformations of set-valued functors. However, stable operations are necessarily homomorphisms of cohomology groups, as the reader is encouraged to check. \begin{thm} Cohomology operations $\tilde{H}^q(-;\pi)\rtarr \tilde{H}^{q+n}(-;\rh)$ are in canonical bijective correspondence with elements of $\tilde{H}^{q+n}(K(\pi,q);\rh)$. \end{thm} \begin{proof} Translate to the represented level, apply the previous corollary, and translate back. \end{proof} This seems very abstract, but it has very concrete consequences. To determine all cohomology operations, we need only compute the cohomology of all Eilenberg-Mac\,Lane spaces. We have described an explicit construction of these spaces as topological Abelian groups in Chapter 16 \S5, and this construction leads to an inductive method of computation. We briefly indicate a key example of how this works, without proofs. \begin{thm} For $n\geq 0$, there are stable cohomology operations $$Sq^n: H^q(X;\bZ_2)\rtarr H^{q+n}(X;\bZ_2),$$ called the Steenrod operations.\index{Steenrod operations} They satisfy the following properties. \begin{enumerate} \item[(i)] $Sq^0$ is the identity operation. \item[(ii)] $Sq^n(x)=x^2$ if $n=\text{\em deg}\,x$ and $Sq^n(x)=0$ if $n> \text{\em deg}\,x$. \item[(iii)] The Cartan formula\index{Cartan formula} holds: $$Sq^n(xy)= \sum_{i+j=n}Sq^i(x)Sq^j(y).$$ \end{enumerate} \end{thm} In fact, the Steenrod operations are uniquely characterized by the stated properties. There are also formulas, called the Adem relations,\index{Adem relations} describing $Sq^iSq^j$, as a linear combination of operations $Sq^{i+j-k}Sq^k$, $2k\leq i$, when $0 \text{\em dim}\,\xi$. \item $w_1(\ga_1)\neq 0$, where $\ga_1$ is the universal line bundle over $\bR P^{\infty}$. \item $w_i(\xi\oplus \epz)= w_i(\xi)$. \item $w_i(\ze\oplus \xi)= \sum_{j=0}^i w_j(\ze)\cup w_{i-j}(\xi)$. \end{enumerate} Every mod $2$ characteristic class for $n$-plane bundles can be written uniquely as a polynomial in the Stiefel-Whitney classes $\sset{w_1,\ldots\!,w_n}$. \end{thm} \begin{thm} For $n\geq 1$, there are elements $w_i\in H^i(BO(n);\bZ_2)$, $i\geq 0$, called the Stiefel-Whitney classes. They satisfy and are uniquely characterized by the following axioms. \begin{enumerate} \item $w_0=1$ and $w_i = 0$ if $i>n$. \item $w_1\neq 0$ when $n=1$. \item $i_n^*(w_i)=w_i$. \item $p_{m,n}^*(w_i) = \sum_{j=0}^i w_j\ten w_{i-j}$. \end{enumerate} The mod $2$ cohomology $H^*(BO(n);\bZ_2)$ is the polynomial algebra $\bZ_2[w_1,\ldots\!,w_n]$. \end{thm} For the uniqueness, suppose given another collection of classes $w'_i$ for all $n\geq 1$ that satisfy the stated properties. Since $BO(1)=\bR P^{\infty}$, $w_1=w'_1$ is the unique non-zero element of $H^1(\bR P^{\infty};\bZ_2)$. Therefore $w_i=w'_i$ for all $i$ when $n=1$, and we assume that this is true for all $mq-n$. Calculation of $w_i(\nu)$ from the Whitney duality formula can lead to a contradiction if $q$ is too small. One calculation is immediate. Since the normal bundle of the standard embedding $S^q\rtarr \bR^{q+1}$ is trivial, $w(S^q)=1$. A manifold is said to be parallelizable\index{parallelizable manifold} if its tangent bundle is trivial. For some manifolds $M$, we can show that $M$ is not parallelizable by showing that one of its Stiefel-Whitney classes is non-zero, but this strategy fails for $M=S^q$. We describe some standard computations in the cohomology of projective spaces that give less trivial examples. Write $\ze_q$ for the canonical line bundle\index{canonical line bundle} over $\bR P^{q}$ in this section. (We called it $\ga_1^{q+1}$ before.) The total space of $\ze_q$ consists of pairs $(x,v)$, where $x$ is a line in $\bR^{q+1}$ and $v$ is a point on that line. This is a subbundle of the trivial $(q+1)$-plane bundle $\epz^{q+1}$, and we write $\ze_q^{\perp}$ for the complementary bundle whose points are pairs $(x,w)$ such that $w$ is orthogonal to the line $x$. Thus $$\ze_q\oplus \ze_q^{\perp}\iso\epz^{q+1}.$$ Write $H^*(\bR P^q;\bZ_2) =\bZ_2[\al]/(\al^{q+1})$, $\deg\al =1$. Thus $\al=w_1(\ze_q)$. Since $\ze_q$ is a line bundle, $w_i(\ze_q)=0$ for $i>1$. The formula $w(\ze_q)\cup w(\ze_q^{\perp})=1$ implies that $$w(\ze_q^{\perp}) = 1+\al +\cdots + \al^{q}.$$ We can describe $\ta(\bR P^q)$ in terms of $\ze_q$. Consider a point $x\in S^q$ and write $(x,v)$ for a typical vector in the tangent plane of $S^q$ at $x$. Then $x$ is orthogonal to $v$ in $\bR^{q+1}$ and $(x,v)$ and $(-x,-v)$ have the same image in $\ta(\bR P^q)$. If $L_x$ is the line through $x$, then this image point determines and is determined by the linear map $f: L_x\rtarr L_x^{\perp}$ that sends $x$ to $v$. Starting from this, it is easy to check that $\ta(\bR P^q)$ is isomorphic to the bundle $\Hom(\ze_q,\ze_q^{\perp})$. As for any line bundle, we have $\Hom(\ze_q,\ze_q)\iso\epz$ since the identity homomorphisms of the fibers specify a cross-section. Again, as for any bundle over a smooth manifold, a choice of Euclidean metric determines an isomorphism $\Hom(\ze_q,\epz)\iso \ze_q$. These facts give the following calculation of $\ta(\bR P^q)\oplus\epz$: \begin{eqnarray*} \ta(\bR P^q)\oplus\epz & \iso & \Hom(\ze_q,\ze_q^{\perp})\oplus\Hom(\ze_q,\ze_q) \\ & \iso & \Hom(\ze_q,\ze_q^{\perp}\oplus \ze_q) \iso \Hom(\ze_q,\epz^{q+1}) \\ & \iso & (q+1)\Hom(\ze_q,\epz)\iso (q+1)\ze_q. \end{eqnarray*} Therefore $$w(\bR P^q) = w((q+1)\ze_q) = w(\ze_q)^{q+1} = (1+\al)^{q+1}=\sum_{0\leq i\leq q} \left(\begin{array}{c}q+1\\i\end{array}\right) \al^i.$$ Explicit computations are obtained by computing mod $2$ binomial coefficients. For example, $w(\bR P^q)=1$ if and only if $q=2^k-1$ for some $k$ (as the reader should check) and therefore $\bR P^q$ can be parallelizable only if $q$ is of this form. If $\bR^{q+1}$ admits a bilinear product without zero divisors, then it is not hard to prove that $\ta(\bR P^{q})\iso \Hom(\ze_q,\ze_q^{\perp})$ admits $q$ linearly independent cross-sections and is therefore trivial. We conclude that $\bR^{q+1}$ can admit such a product only if $q+1=2^k$ for some $k$. The real numbers, complex numbers, quaternions, and Cayley numbers show that there is such a product for $q+1=1$, $2$, $4$, and $8$. As we shall explain in the next chapter, these are in fact the only $q$ for which $\bR^{q+1}$ admits such a product. While the calculation of $w(\bR P^q)$ just given is quite special, there is a remarkable general recipe, called the ``Wu formula,'' for the computation of $w(M)$ in terms of Poincar\'e duality and the Steenrod operations in $H^*(M;\bZ_2)$. In analogy with $w(M)$, we define the total Steenrod square of an element $x$ by $Sq(x)=\sum_i Sq^i(x)$.\index{total Steenrod operation} \begin{thm}[Wu formula]\index{Wu formula} Let $M$ be a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\bZ_2)$. Then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=\sum v_i\in H^{**}(M;\bZ_2)$ is the unique cohomology class such that $$\langle v\cup x,z\rangle = \langle Sq(x),z \rangle $$ for all $x\in H^*(M;\bZ_2)$. Thus, for $k\geq 0$, $v_k\cup x = Sq^k(x)$ for all $x\in H^{n-k}(M;\bZ_2)$, and $$w_k(M)=\sum_{i+j=k}Sq^i(v_j).$$ \end{thm} Here the existence and uniqueness of $v$ is an easy exercise from the Poincar\'e duality theorem. The basic reason that such a formula holds is that the Stiefel-Whitney classes can be defined in terms of the Steenrod operations, as we shall see shortly. The Wu formula implies that the Stiefel-Whitney classes are homotopy invariant: if $f:M\rtarr M'$ is a homotopy equivalence between smooth closed $n$-manifolds, then $f^*: H^*(M';\bZ_2)\rtarr H^*(M;\bZ_2)$ satisfies $f^*(w(M'))=w(M)$. In fact, the conclusion holds for any map $f$, not necessarily a homotopy equivalence, that induces an isomorphism in mod $2$ cohomology. Since the tangent bundle of $M$ depends on its smooth structure, this is rather surprising. \section{Characteristic numbers of manifolds} Characteristic classes determine important numerical invariants of manifolds, called their characteristic numbers. \begin{defn} Let $M$ be a smooth closed $R$-oriented $n$-manifold with fundamental class $z\in H_n(M;R)$. For a characteristic class $c$ of degree $n$, define the tangential characteristic number\index{characteristic class}\index{characteristic number!tangential} \index{characteristic number!normal} $c[M]\in R$ by $c[M] = \langle c(\ta(M)),z \rangle$. Similarly, define the normal characteristic number $c[\nu(M)]$ by $c[\nu(M)] = \langle c(\nu(M)),z \rangle$, where $\nu(M)$ is the normal bundle associated to an embedding of $M$ in $\bR^q$ for $q$ sufficiently large. (These numbers are well defined because any two embeddings of $M$ in $\bR^q$ for large $q$ are isotopic and have equivalent normal bundles.) \end{defn} In particular, if $r_i$ are integers such that $\sum ir_i=n$, then the monomial $w_1^{r_1}\cdots w_n^{r_n}$ is a characteristic class of degree $n$, and all mod $2$ characteristic classes of degree $n$ are linear combinations of these. Different manifolds can have the same Stiefel-Whitney numbers.\index{Stiefel-Whitney numbers} In fact, we have the following observation. \begin{lem} If $M$ is the boundary of a smooth compact $(n+1)$-manifold $W$, then all tangential Stiefel-Whitney numbers of $M$ are zero. \end{lem} \begin{proof} Using a smooth tubular neighborhood, we see that there is an inward-pointing normal vector field along $M$ that spans a trivial bundle $\epz$ such that $$\ta(W)|_M\iso \ta(M)\oplus \epz.$$ Therefore, if $i:M\rtarr W$ is the inclusion, then $i^*(w_j(W))=w_j(M)$. Let $f$ be a polynomial in the $w_j$ of degree $n$. Recall that the fundamental class of $M$ is $\pa z$, where $z\in H_{n+1}(W,M)$ is the fundamental class of the pair $(W,M)$. We have $$\langle f(M),\pa z \rangle = \langle i^*f(W),\pa z \rangle = \langle f(W), i_*\pa z \rangle = 0$$ since $i_*\pa = 0$ by the long exact homology sequence of the pair. \end{proof} \begin{lem} All tangential Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!tangential} of a smooth closed manifold $M$ are zero if and only if all normal Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!normal} of $M$ are zero. \end{lem} \begin{proof} The Whitney duality formula implies that every $w_i(M)$ is a polynomial in the $w_i(\nu(M))$ and every $w_i(\nu(M))$ is a polynomial in the $w_i(M)$. \end{proof} We shall explain the following amazing result of Thom in the last chapter. \begin{thm}[Thom] If $M$ is a smooth closed $n$-manifold all of whose normal Stiefel-Whitney numbers are zero, then $M$ is the boundary of a smooth $(n+1)$-manifold. \end{thm} Thus we need only compute the Stiefel-Whitney numbers of $M$ to determine whether or not it is a boundary. By Wu's formula, the computation only requires knowledge of the mod $2$ cohomology of $M$, with its Steenrod operations. In practice, it might be fiendishly difficult to actually construct a manifold with boundary $M$ geometrically. \section{Thom spaces and the Thom isomorphism theorem} There are several ways to construct the Stiefel-Whitney classes. The most illuminating one depends on a simple, but fundamentally important, construction on vector bundles, namely their ``Thom spaces.'' This construction will also be at the heart of the proof of Thom's theorem in the last chapter. \begin{defn} Let $\xi: E\rtarr B$ be an $n$-plane bundle. Apply one-point compactification to each fiber of $\xi$ to obtain a new bundle $Sph(E)$\index{Sph(E)@$Sph(E)$} over $B$ whose fibers are spheres $S^n$ with given basepoints, namely the points at $\infty$. These basepoints specify a cross-section $B\rtarr Sph(E)$. Define the Thom space\index{Thom space} $T\xi$ to be the quotient space $T(\xi)= Sph(E)/B$. That is, $T(\xi)$ is obtained from $E$ by applying fiberwise one-point compactification and then identifying all of the points at $\infty$ to a single basepoint (denoted $\infty$). Observe that this construction is functorial with respect to maps of vector bundles. \end{defn} \begin{rem} If we give the bundle $\xi$ a Euclidean metric and let $D(E)$ and $S(E)$ denote its unit disk bundle and unit sphere bundle, then there is an evident homeomorphism between $T\xi$ and the quotient space $D(E)/S(E)$. In turn, $D(E)/S(E)$ is homotopy equivalent to the cofiber of the inclusion $S(E)\rtarr D(E)$ and therefore to the cofiber of the projection $S(E)\rtarr B$. \end{rem} If the bundle $\xi$ is trivial, so that $E=B\times \bR^n$, then $ Sph(E)=B\times S^n$. Quotienting out $B$ amounts to the same thing as giving $B$ a disjoint basepoint and then forming the smash product $B_+\sma S^n$. That is, in this case the Thom complex is $\SI^nB_+$. Therefore, for any cohomology theory $k^*$, $$k^q(B)=\tilde{k}^q(B_+) \iso \tilde{k}^{n+q}(T\xi).$$ There is a conceptual way of realizing this isomorphism. For any $n$-plane bundle $\xi: E\rtarr B$, we have a projection $\xi: Sph(E)\rtarr B$ and a quotient map $\pi: Sph(E)\rtarr T\xi$. We can compose their product with the diagonal map of $Sph(E)$ to obtain a composite map $$ Sph(E)\rtarr Sph(E)\times Sph(E) \rtarr B\times T\xi.$$ This sends all points at $\infty$ to points of $B\times \sset{\infty}$. Therefore it factors through a map $$ \DE: T\xi\rtarr B_+\sma T\xi,$$ which is called the ``Thom diagonal.''\index{Thom diagonal} For a commutative ring $R$, we can use $\DE$ to define a cup product $$ H^p(B;R)\ten \tilde{H}^q(T\xi;R) \rtarr \tilde{H}^{p+q}(T\xi;R).$$ When the bundle $\xi$ is trivial, we let $\mu\in \tilde{H}^n(B_+\sma S^n;R)$ be the suspension of the identity element $1\in H^0(B;R)$, and we find that $x\rtarr x\cup \mu$ specifies the suspension isomorphism $H^q(B;R)\iso \tilde{H}^{n+q}(B_+\sma S^n;R) = \tilde{H}^{n+q}(T\xi;R)$. Now consider a general bundle $\xi$. On neighborhoods $U$ of $B$ over which $\xi$ is trivial, we have $H^q(U;R)\iso \tilde{H}^{n+q}(T(\xi|_U);R)$. The isomorphism depends on the trivialization $\ph_U: U\times \bR^n\rtarr \xi^{-1}(U)$. It is natural to ask if these isomorphisms patch together to give a global isomorphism $H^q(B_+)\rtarr \tilde{H}^{n+q}(T\xi)$. This should look very similar to the problem of patching local fundamental classes to obtain a global one; that is, it looks like a question of orientation. This leads to the following definition and theorem. For a point $b\in B$, let $S^n_b$ be the one-point compactification of the fiber $\xi^{-1}(b)$; since $S^n_b$ is the Thom space of $\xi|_b$, we have a canonical map $i_b: S^n_b\rtarr T\xi$. \begin{defn} Let $\xi: E\rtarr B$ be an $n$-plane bundle. An $R$-orientation,\index{Rorientation@$R$-orientation} or Thom class,\index{Thom class} of $\xi$ is an element $\mu\in \tilde{H}^n(T\xi;R)$ such that, for every point $b\in B$, $i_b^*(\mu)$ is a generator of the free $R$-module $\tilde{H}^n(S^n_b)$. \end{defn} We leave it as an instructive exercise to verify that an $R$-orientation of a closed $n$-manifold $M$ determines and is determined by an $R$-orientation of its tangent bundle $\ta(M)$. \begin{thm}[Thom isomorphism theorem]\index{Thom isomorphism} Let $\mu\in \tilde{H}^n(T\xi;R)$ be a Thom class for an $n$-plane bundle $\xi: E\rtarr B$. Define $$\PH: H^q(B;R)\rtarr \tilde{H}^{n+q}(T\xi;R)$$ by $\PH (x)=x\cup \mu$. Then $\PH$ is an isomorphism. \end{thm} \begin{proof}[Sketch Proof] When $R$ is a field, this can be proved by an inductive Mayer-Vietoris sequence argument. To exploit inverse images of open subsets of $B$, it is convenient to observe that, by easy homotopy and excision arguments, $$\tilde{H}^*(T\xi)\iso H^*(Sph(E),B)\iso H^*(Sph(E),Sph(E)_0)\iso H^*(E,E_0),$$ where $E_0$ and $Sph(E)_0$ are the subspaces of $E$ and $Sph(E)$ obtained by deleting $\sset{0}$ from each fiber. Use of a field ensures that the cohomology of the relevant direct limits is the inverse limit of the cohomologies. An alternative argument that works for general $R$ can be obtained by first showing that one can assume that $B$ is a CW complex, by replacing $\xi$ by its pullback along a CW approximation of $B$, and then proceeding by induction over the restrictions of $\xi$ to the skeleta of $B$; one point is that the restriction of $\xi$ to any cell is trivial and another is that the cohomology of $B$ is the inverse limit of the cohomologies of its skeleta. However, much the best proof from the point of view of anyone seriously interested in algebraic topology is to apply the Serre spectral sequence of the bundle $Sph(E)$. The Serre spectral sequence\index{Serre spectral sequence} is a device for computing the cohomology of the total space $E$ of a fibration from the cohomologies of its base $B$ and fiber $F$. It measures the cohomological deviation of $H^*(E)$ from $H^*(B)\ten H^*(F)$. In the present situation, the existence of a Thom class ensures that there is no deviation for the sphere bundle $Sph(E)\rtarr B$, so that $$H^*(Sph(E);R)\iso H^*(B;R)\ten H^*(S^n;R).$$ The section given by the points at $\infty$ induces an isomorphism of $H^*(B;R)\ten H^0(S^n;R)$ with $H^*(B;R)$, and the quotient map $Sph(E)\rtarr T\xi$ induces an isomorphism of $\tilde{H}^*(T\xi;R)$ with $H^*(B;R)\ten H^n(S^n;R)$. \end{proof} Just as in orientation theory for manifolds, the question of orientability depends on the structure of the units of the ring $R$, and this leads to the following conclusion. \begin{prop} Every vector bundle admits a unique $\bZ_2$-orientation. \end{prop} This can be proved along with the Thom isomorphism theorem by a Mayer-Vietoris argument. \section{The construction of the Stiefel-Whitney classes} We indicate two constructions of the Stiefel-Whitney classes. Each has distinct advantages over the other. First, taking the characteristic class point of view, we define the Stiefel-Whitney classes\index{Stiefel-Whitney classes} in terms of the Steenrod operations by setting $$w_i(\xi) = \PH^{-1}Sq^i\PH(1) = \PH^{-1}Sq^i\mu.$$ Naturality is obvious. Axiom 1 is immediate from the relations $Sq^0=\id$ and $Sq^i(x)=0$ if $i> \deg\,x$. For axiom 2, we use the following observation. \begin{lem} There is a homotopy equivalence $j: \bR P^{\infty}\rtarr T\ga_1$. \end{lem} \begin{proof} $T\ga_1$ is homeomorphic to $D(\ga_1)/S(\ga_1)$. Here $S(\ga_1)$ is the infinite sphere $S^{\infty}$, which is the universal cover of $\bR P^{\infty}$ and is therefore contractible. The zero section $\bR P^{\infty}\rtarr D(\ga_1)$ and the quotient map $D(\ga_1)\rtarr T\ga_1$ are homotopy equivalences, and their composite is the required homotopy equivalence $j$. \end{proof} Since $Sq^1(x)=x^2$ if $\deg\,x=1$, the lemma implies that $Sq^1$ is non-zero on the Thom class of $\ga_1$, verifying axiom 2. For axiom 3, we easily check that $T(\xi\oplus\epz)\iso \SI T(\xi)$ for any vector bundle $\xi$ and that the Thom class of $\xi\oplus\epz$ is the suspension of the Thom class of $\xi$. Thus axiom 3 follows from the stability of the Steenrod operations. For axiom 4, we easily check that, for any vector bundles $\ze$ and $\xi$, $T(\ze\times \xi)\iso T\ze\sma T\xi$ and the Thom class of $\ze\times \xi$ is the tensor product of the Thom classes of $\ze$ and $\xi$. Interpreting the Cartan formula for the Steenrod operations externally in the cohomology of products and therefore of smash products, we see that it implies axiom 4. That is, the properties that axiomatize the Steenrod operations directly imply the properties that axiomatize the Stiefel-Whitney classes. We next take the classifying space point of view. As we shall explain in \S8, passage from topological groups to their classifying spaces is a product-preserving functor, at least up to homotopy. We may embed $(\bZ_2)^n = O(1)^n$ in $O(n)$ as the subgroup of diagonal matrices. The classifying space $BO(1)$ is $\bR P^{\infty}$, and we obtain a map $$\om: (\bR P^{\infty})^n \htp B(O(1)^n) \rtarr BO(n) $$ upon passage to classifying spaces. The symmetric group $\SI_n$ is contained in $O(n)$ as the subgroup of permutation matrices, and the diagonal subgroup $O(1)^n$ is closed under conjugation by symmetric matrices. Application of the classifying space functor to conjugation by permutation matrices induces the corresponding permutation of the factors of $BO(1)^n$, and it induces the identity map on $BO(n)$. Indeed, up to homotopy, inner conjugation by an element of $G$ induces the identity map on $BG$ for any topological group $G$. By the K\"unneth theorem, we see that $$H^*((\bR P^{\infty})^n;\bZ_2) = \ten_{i=1}^n H^*(\bR P^{\infty};\bZ_2) =\bZ_2[\al_1,\ldots\!,\al_n],$$ where the generators $\al_i$ are of degree one. The symmetric group $\SI_n$ acts on this cohomology ring by permuting the variables $\al_i$. The subring $H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$ of elements invariant under the action is the polynomial algebra on the elementary symmetric functions $\si_i$, $1\leq i\leq n$, in the variables $\al_i$. Here $$\si_i = \textstyle{\sum} \al_{j_1}\cdots\al_{j_i},\ \ 1\leq j_1 < \cdots < j_n,$$ has degree $i$. The induced map $\om^*: H^*(BO(n);\bZ_2)\rtarr H^*((\bR P^{\infty})^n;\bZ_2)$ takes values in $H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$. We shall give a general reason why this is so in \S8. The resulting map $$\om^*: H^*(BO(n);\bZ_2)\rtarr H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$$ is a ring homomorphism between polynomial algebras on generators of the same degrees. It turns out to be a monomorphism and therefore an isomorphism. We redefine the Stiefel-Whitney classes by letting $w_i$ be the unique element such that $\om^*(w_i)=\si_i$ for $1\leq i\leq n$ and defining $w_0=1$ and $w_i=0$ for $i>n$. Then axioms 1 and 2 for the Stiefel-Whitney classes are obvious, and we derive axioms 3 and 4 from algebraic properties of elementary symmetric functions. One advantage of this approach is that, since we know the Steenrod operations on $H^*(\bR P^{\infty};\bZ_2)$ and can read them off on $H^*((\bR P^{\infty})^n;\bZ_2)$ by the Cartan formula, it leads to a purely algebraic calculation of the Steenrod operations in $H^*(BO(n);\bZ_2)$. Explicitly, the following ``Wu formula''\index{Wu formula} holds: $$ Sq^i(w_j) = \sum_{t=0}^i\left(\begin{array}{c}j+t-i-1\\t\end{array}\right) w_{i-t}w_{j+t}.$$ \section{Chern, Pontryagin, and Euler classes} The theory of the previous sections extends appropriately to complex vector bundles and to oriented real vector bundles. The proof of the classification theorem for complex $n$-plane bundles works in exactly the same way as for real $n$-plane bundles, using complex Grassmann varieties. For oriented real $n$-plane bundles, we use the Grassmann varieties\index{Grassmann variety!of oriented $n$-planes} of oriented $n$-planes, the points of which are planes $x$ together with a chosen orientation. In fact, the fundamental groups of the real Grassmann varieties are $\bZ_2$, and their universal covers are their orientation covers. These covers are the oriented Grassmann varieties $\tilde{G}_n(\bR^q)$. We write $BU(n) = G_n(\bC^{\infty})$\index{BUn@$BU(n)$} and $BSO(n) = \tilde{G}_n(\bR^{\infty})$,\index{BSOn@$BSO(n)$} and we construct universal complex $n$-plane bundles\index{universal n-plane bundle@universal $n$-plane bundle!complex} \index{universal n-plane bundle@universal $n$-plane bundle!oriented} $\ga_n: EU_n\rtarr BU(n)$ and oriented $n$-plane bundles $\tilde{\ga}_n: \tilde{E}_n\rtarr BSO(n)$ as in the first section. Let $\sE U_n(B)$\index{EUkn(-)@$\sE U_n(B)$} denote the set of equivalence classes of complex $n$-plane bundles over $B$ and let $\tilde{\sE}_n(B)$\index{EanBa@$\tilde{\sE}_n(B)$} denote the set of equivalence classes of oriented real $n$-plane bundles over $B$; it is required that bundle maps $(g,f)$ be orientation preserving, in the sense that the induced map of Thom spaces carries the orientation of the target bundle to the orientation of the source bundle. The universal bundle $\tilde{\ga_n}$ has a canonical orientation which determines an orientation on $f^*\tilde{E}_n$ for any map $f: B\rtarr BSO(n)$. \begin{thm}\index{classification theorem!for complex $n$-plane bundles} The natural transformation $\PH: [-,BU(n)]\rtarr \sE U_n(-)$ obtained by sending the homotopy class of a map $f: B\rtarr BU(n)$ to the equivalence class of the $n$-plane bundle $f^*EU_n$ is a natural isomorphism of functors. \end{thm} \begin{thm}\index{classification theorem!for oriented $n$-plane bundles} The natural transformation $\PH: [-,BSO(n)]\rtarr \tilde{\sE}_n(-)$ obtained by sending the homotopy class of a map $f: B\rtarr BSO(n)$ to the equivalence class of the oriented $n$-plane bundle $f^*\tilde{E}_n$ is a natural isomorphism of functors. \end{thm} The definition of characteristic classes for complex $n$-plane bundles and for oriented real $n$-plane bundles in a cohomology theory $k^*$ is the same as for real $n$-plane bundles, and the Yoneda lemma applies. \begin{lem} Evaluation on $\ga_n$ specifies a canonical bijection between characteristic classes of complex $n$-plane bundles and elements of $k^*(BU(n))$. \end{lem} \begin{lem} Evaluation on $\tilde{\ga}_n$ specifies a canonical bijection between characteristic classes of oriented $n$-plane bundles and elements of $k^*(BSO(n))$. \end{lem} Clearly we have a $2$-fold cover $\pi_n: BSO(n)\rtarr BO(n)$. The mod $2$ characteristic classes for oriented $n$-plane bundles are as one might expect from this. Continue to write $w_i$ for $\pi^*(w_i)\in H^i(BSO(n);\bZ_2)$; here $w_1=0$ since $BSO(n)$ is simply connected. \begin{thm} $H^*(BSO(n);\bZ_2) \iso \bZ_2[w_2,\ldots\!,w_n]$. \end{thm} If we regard a complex $n$-plane bundle as a real $2n$-plane bundle, then the complex structure induces a canonical orientation. By the Yoneda lemma, the resulting natural transformation $r: \sE U_n(-)\rtarr \tilde{\sE}_n(-)$ is represented by a map $r: BU(n)\rtarr BSO(2n)$. Explicitly, ignoring its complex structure, we may identify $\bC^{\infty}$ with $\bR^{\infty}\oplus\bR^{\infty}\iso \bR^{\infty}$ and so regard a complex $n$-plane in $\bC^{\infty}$ as an oriented $2n$-plane in $\bR^{\infty}$. Similarly, we may complexify real bundles fiberwise and so obtain a natural transformation $c: \sE_n(-)\rtarr \sE U_n(-)$. It is represented by a map $c: BO(n)\rtarr BU(n)$. Explicitly, identifying $\bC^{\infty}$ with $\bR^{\infty}\ten_{\bR}{\bC}$, we may complexify an $n$-plane in $\bR^{\infty}$ to obtain an $n$-plane in $\bC^{\infty}$. The Thom space\index{Thom space!of a complex bundle} of a complex or oriented real vector bundle is the Thom space of its underlying real vector bundle. We obtain characteristic classes in cohomology with any coefficients by applying cohomology operations to Thom classes, but it is rarely the case that the resulting characteristic classes generate all characteristic classes: the cases $H^*(BO(n);\bZ_2)$ and $H^*(BSO(n);\bZ_2)$ are exceptional. Characteristic classes constructed in this fashion satisfy homotopy invariance properties that fail for general characteristic classes. In the complex case, with integral coefficients, we have a parallel to our second approach to Stiefel-Whitney classes that leads to a description of $H^*(BU(n);\bZ)$ in terms of Chern classes. We may embed $(S^1)^n = U(1)^n$ in $U(n)$ as the subgroup of diagonal matrices. The classifying space $BU(1)$ is $\bC P^{\infty}$, and we obtain a map $$\om: (\bC P^{\infty})^n \htp B(U(1)^n) \rtarr BU(n) $$ upon passage to classifying spaces. The symmetric group $\SI_n$ is contained in $U(n)$ as the subgroup of permutation matrices, and the diagonal subgroup $U(1)^n$ is closed under conjugation by symmetric matrices. Application of the classifying space functor to conjugation by permutation matrices induces the corresponding permutation of the factors of $BU(1)^n$, and it induces the identity map on $BU(n)$. By the K\"unneth theorem, we see that $$H^*((\bC P^{\infty})^n;\bZ) = \ten_{i=1}^n H^*(\bC P^{\infty};\bZ) =\bZ[\be_1,\ldots\!,\be_n],$$ where the generators $\be_i$ are of degree two. The symmetric group $\SI_n$ acts on this cohomology ring by permuting the variables $\be_i$. The subring $H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$ of elements invariant under the action is the polynomial algebra on the elementary symmetric functions $\si_i$, $1\leq i\leq n$, in the variables $\be_i$. Here $$\si_i = \textstyle{\sum} \be_{j_1}\cdots\be_{j_i},\ \ 1\leq j_1 < \cdots < j_n,$$ has degree $2i$. The induced map $\om^*: H^*(BU(n);\bZ)\rtarr H^*((\bC P^{\infty})^n;\bZ)$ takes values in $H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$. The resulting map $$\om^*: H^*(BU(n);\bZ)\rtarr H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$$ is a ring homomorphism between polynomial algebras on generators of the same degrees. It turns out to be a monomorphism and thus an isomorphism when tensored with any field, and it is therefore an isomorphism. We define the Chern classes by letting $c_i$, $1\leq i\leq n$, be the unique element such that $\om^*(c_i)=\si_i$. \begin{thm} For $n\geq 1$, there are elements $c_i\in H^{2i}(BU(n);\bZ)$, $i\geq 0$, called the Chern classes.\index{Chern classes} They satisfy and are uniquely characterized by the following axioms. \begin{enumerate} \item $c_0=1$ and $c_i = 0$ if $i>n$. \item $c_1$ is the canonical generator of $H^2(BU(1);\bZ)$ when $n=1$. \item $i_n^*(c_i)=c_i$. \item $p_{m,n}^*(c_i) = \sum_{j=0}^i c_j\ten c_{i-j}$. \end{enumerate} The integral cohomology $H^*(BU(n);\bZ)$ is the polynomial algebra $\bZ[c_1,\ldots\!,c_n]$. \end{thm} Here we take axiom 1 as a definition and we interpret axiom 2 as meaning that $c_1$ corresponds to the identity map of $\bC P^{\infty}$ under the canonical identification of $[\bC P^{\infty},\bC P^{\infty}]$ with $H^2(\bC P^{\infty};\bZ)$. Axioms 3 and 4 can be read off from algebraic properties of elementary symmetric functions. The theorem admits an immediate interpretation in terms of characteristic classes. Observe that, since $H^*(BU(n);\bZ)$ is a free Abelian group, the theorem remains true precisely as stated with $\bZ$ replaced by any other commutative ring of coefficients $R$. We continue to write $c_i$ for the image of $c_i$ in $H^*(BU(n);R)$ under the homomorphism induced by the unit $\bZ\rtarr R$ of the ring $R$. The reader deserves to be warned about a basic inconsistency in the literature. \begin{rem} With the discussion above, $c_1(\ga_1^{n+1})$ is the canonical generator of $H^2(\bC P^n;\bZ)$, where $\ga_1^{n+1}$ is the canonical line bundle \index{canonical line bundle} of lines in $\bC^{n+1}$ and points on the line. This is the standard convention in algebraic topology. In algebraic geometry, it is more usual to define Chern classes so that the first Chern class of the dual of $\ga_1^{n+1}$ is the canonical generator of $H^2(\bC P^n;\bZ)$. With this convention, the $n$th Chern class would be $(-1)^nc_n$. It is often unclear in the literature which convention is being followed. \end{rem} Turning to oriented real vector bundles, we define the Pontryagin and Euler classes as follows, taking cohomology with coefficients in any commutative ring $R$. \begin{defn} Define the Pontryagin classes\index{Pontryagin classes} $p_i\in H^{4i}(BO(n);R)$ by $$p_i = (-1)^ic^*(c_{2i}),$$ $c^*: H^{4i}(BU(n);R)\rtarr H^{4i}(BO(n);R)$; also write $p_i$ for $\pi_n^*(p_i)\in H^{4i}(BSO(n);R)$. \end{defn} \begin{defn} Define the Euler class\index{Euler class} $e(\xi)\in H^n(B;R)$ of an $R$-oriented $n$-plane bundle $\xi$ over the base space $B$ by $e(\xi)=\PH^{-1}\mu^2$, where $\mu\in H^n(T\xi;R)$ is the Thom class. Giving the universal oriented $n$-plane bundle over $BSO(n)$ the $R$-orientation induced by its integral orientation, this defines the Euler class $e\in H^n(BSO(n);R)$. \end{defn} If $n$ is odd, then $2\mu^2=0$ and thus $2e=0$. If $R=\bZ_2$, then $Sq^n(\mu)=\mu^2$ and thus $e=w_n$. The name ``Euler class'' is justified by the following classical result, which well illustrates the kind of information that characteristic numbers can encode.\footnote{See Corollary 11.12 of Milnor and Stasheff {\em Characteristic Classes} for a proof.} \begin{thm} If $M$ is a smooth closed oriented manifold, then the characteristic number $e[M]=\langle e(\ta(M)),z\rangle\in \bZ$ is the Euler characteristic\index{Euler characteristic} of $M$. \end{thm} The evident inclusion $T^n \iso SO(2)^n\rtarr SO(2n)$ is a maximal torus, and it induces a map $BT^n\rtarr BSO(2n)$. A calculation shows that $e$ restricts to the $n$th elementary symmetric polynomial $\be_1\cdots\be_n$. The cited inclusion factors through the homomorphism $U(n)\rtarr SO(2n)$, hence $BT^n\rtarr BSO(2n)$ factors through $r: BU(n)\rtarr BSO(2n)$. This implies another basic fact about the Euler class. \begin{prop} $r^*: H^*(BSO(2n);\bZ)\rtarr H^*(BU(n);\bZ)$ sends $e$ to $c_n$. \end{prop} The presence of $2$-torsion makes the description of the integral cohomology rings of $BO(n)$ and $BSO(n)$ quite complicated, and these rings are almost never used in applications. Rather, one uses the mod $2$ cohomology rings and the following description of the cohomology rings that result by elimination of $2$-torsion. \begin{thm} Take coefficients in a ring $R$ in which $2$ is a unit. Then $$H^*(BO(2n)) \iso H^*(BO(2n+1))\iso H^*(BSO(2n+1)) \iso R[p_1,\ldots\!,p_n]$$ and $$H^*(BSO(2n))\iso R[p_1,\ldots\!,p_{n-1},e], \, \, \text{with}\, \, e^2=p_n.$$ \end{thm} \section{A glimpse at the general theory} We should place the theory of vector bundles in a more general context. We have written $BO(n)$, $BU(n)$, and $BSO(n)$ for certain ``classifying spaces'' in this chapter, but we defined a classifying space $BG$ for any topological group $G$ in Chapter 16 \S5. In fact, the spaces here are homotopy equivalent to the spaces of the same name that we defined there, and we here explain why. Consider bundles $\xi: Y\rtarr B$ with fiber $G$. For spaces $U$ in a numerable open cover $\sO$ of $B$, there are homeomorphisms $\ph: U\times G\rtarr p^{-1}(U)$ such that $p\com \ph= \pi_1$. We say that $Y$ is a principal $G$-bundle \index{principal G-bundle@principal $G$-bundle} if $Y$ has a free right action by $G$, $B$ is the orbit space $Y/G$, $\xi$ is the quotient map, and the $\ph$ are maps of right $G$-spaces. We say that $\xi: Y\rtarr B$ is a universal principal $G$-bundle\index{universal principal G-bundle@universal principal $G$-bundle} if $Y$ is a contractible space. In particular, for any topological group $G$ whose identity element is a nondegenerate basepoint, such as any Lie group $G$, the map $p: EG\rtarr BG$ constructed in Chapter 16 \S5 is a universal principal $G$-bundle. The classification theorem below implies that the base spaces of any two universal principal $G$-bundles are homotopy equivalent, and it is usual to write $BG$ for any space\index{classifying space} in this homotopy type. Observe that the long exact sequence of homotopy groups of a universal principal $G$-bundle gives isomorphisms $\pi_q(BG)\iso \pi_{q-1}(G)$ for $q\geq 1$. We have implicitly constructed other examples of universal principal $G$-bundles when $G$ is $O(n)$, $U(n)$, or $SO(n)$. To see this, consider $V_n(\bR^q)$. Write $\bR^q=\bR^n\times \bR^{q-n}$ and note that this fixes embeddings of $O(n)$ and $O(q-n)$ in the orthogonal group $O(q)$. Of course, $O(q)$ acts on vectors in $\bR^q$ and thus on $n$-frames. Consider the fixed $n$-frame $x_0=\sset{e_1,\ldots\!,e_n}$. Any other $n$-frame can be obtained from this one by the action of an element of $O(q)$, and the isotropy group of $x_0$ is $O(q-n)$. Thus the action of $O(q)$ is transitive, and evaluation on $x_0$ induces a homeomorphism $O(q)/O(q-n) \rtarr V_n(\bR^q)$ of $O(q)$-spaces. The action of $O(n)\subset O(q)$ is free, and passage to orbits gives a homeomorphism $O(q)/O(n)\times O(q-n) \rtarr G_n(\bR^q)$. It is intuitively clear and not hard to prove that the colimit over $q$ of the inclusions $O(q-n)\rtarr O(q)$ is a homotopy equivalence and that this implies the contractibility of $V_n(\bR^{\infty})$. We deduce that $V_n(\bR^{\infty})$ is a universal principal $O(n)$-bundle. We have analogous universal principal $U(n)$-bundles and $SO(n)$-bundles. There is a classification theorem\index{classification theorem!for principal $G$-bundles} for principal $G$-bundles. Let $\sP G(B)$\index{PG(B)@$\sP G(B)$} denote the set of equivalence classes of principal $G$-bundles over $B$, where two principal $G$-bundles over $B$ are equivalent if there is a $G$-homeomorphism over $B$ between them. Via pullback of bundles, this is a contravariant set-valued functor on the homotopy category of spaces. \begin{thm} Let $\ga: Y\rtarr Y/G$ be any universal principal $G$-bundle. The natural transformation $\PH: [-,Y/G]\rtarr \sP G(-)$ obtained by sending the homotopy class of a map $f: B\rtarr Y/G$ to the equivalence class of the principal $G$-bundle $f^*Y$ is a natural isomorphism of functors. \end{thm} Now let $F$ be any space on which $G$ acts effectively from the left. Here an action is effective\index{effective group action} if $gf=f$ for every $f\in F$ implies $g=e$. For a principal $G$-bundle $Y$, let $G$ act on $Y\times F$ by $g(y,f)=(yg^{-1},gf)$ and let $Y\times_G F$ be the orbit space $(Y\times F)/G$. With the correct formal definition of a fiber bundle with group $G$ and fiber $F$, every such fiber bundle $p: E \rtarr B$ is equivalent to one of the form $Y\times_G F\rtarr Y/G\iso B$ for some principal $G$-bundle $Y$ over $B$; moreover $Y$ is uniquely determined up to equivalence. In fact, the ``associated principal $G$-bundle''\index{associated principal $G$-bundle} $Y$ can be constructed as the function space of all maps $\ps:F\rtarr E$ such that $\ps$ is an admissible homeomorphism onto some fiber $F_b=p^{-1}(b)$. Here admissibility means that the composite of $\ps$ with the homeomorphism $F_b\rtarr F$ determined by a coordinate chart $\ph: U\times F\overto{\iso} p^{-1}(U)$, $b\in U$, coincides with action by some element of $G$. The left action of $G$ on $F$ induces a right action of $G$ on $Y$; this action is free because the given action on $F$ is effective. The projection $Y\rtarr B$ sends $\ps$ to $b$ when $\ps: F\overto{\iso} F_b$, and it factors through a homeomorphism $Y/G\rtarr B$. $Y$ inherits local triviality from $p$, and the evaluation map $Y\times F\rtarr E$ induces an equivalence of bundles $Y\times_G F\rtarr E$. We conclude that, for any $F$, $\sP G(B)$ is naturally isomorphic to the set of equivalence classes of bundles with group $G$ and fiber $F$ over $B$. Fiber bundles with group $O(n)$ and fiber $\bR^n$ are real $n$-plane bundles, fiber bundles with group $U(n)$ and fiber $\bC^n$ are complex $n$-plane bundles, and fiber bundles with group $SO(n)$ and fiber $\bR^n$ are oriented real $n$-plane bundles. Thus the classification theorems of the previous sections could all be rederived as special cases of the general classification theorem for principal $G$-bundles stated in this section. In our discussion of Stiefel-Whitney and Chern classes, we used that passage to classifying spaces is a product-preserving functor, at least up to homotopy. For the functoriality, if $f: G\rtarr H$ is a homomorphism of topological groups, then consideration of the way bundles are constructed by gluing together coordinate charts shows that a principal $G$-bundle $\xi: Y\rtarr B$ naturally gives rise to a principal $H$-bundle $f_*Y\rtarr B$. This construction is represented on the classifying space level by a map $Bf: BG\rtarr BH$. In fact, if $EG\rtarr BG$ and $EH\rtarr BH$ are universal principal bundles, then any map $\tilde{f}: EG\rtarr EH$ such that $\tilde{f}(xg)=\tilde{f}(x)f(g)$ for all $x\in EG$ and $g\in G$ induces a map in the homotopy class $Bf$ on passage to orbits. For example, if $f: G\rtarr G$ is given by conjugation by $\ga\in G$, $f(g) = \ga^{-1}g\ga$, then $\tilde{f}(x) = x\ga$ satisfies this equivariance property and therefore $Bf$ is homotopic to the identity. This explains why inner conjugations induce the identity map on passage to classifying spaces, as we used in our discussion of Stiefel-Whitney and Chern classes. If $EG\rtarr BG$ and $EG'\rtarr BG'$ are universal principal $G$ and $G'$ bundles, then $EG\times EG'$ is a contractible space with a free action by $G\times G'$. The orbit space is $BG\times BG'$, and this shows that $BG\times BG'$ is a choice for the classifying space $B(G\times G')$ and is therefore homotopy equivalent to any other choice. The explicit construction of $BG$ given in Chapter 16 \S5 is functorial in $G$ on the point-set level and not just up to homotopy, and it is product preserving in the strong sense that the projections induce a homeomorphism $B(H\times G)\iso BH\times BG$. \vspace{.1in} \begin{center} PROBLEMS \end{center} \begin{enumerate} \item Verify that $w(\bR P^q)=1$ if and only if $q=2^k-1$ for some $k$. \item Prove that $\bR P^{2^k}$ cannot immerse in $\bR^{2^{k+1}-2}$. (By the Whitney embedding theorem, any smooth closed $n$-manifold immerses in $\bR^{2n-1}$, so this is a best possible non-immersion result.) \item Prove that all tangential Stiefel-Whitney numbers of $\bR P^{q}$ are zero if and only if $q$ is odd. \item* Try to construct a smooth compact manifold whose boundary is $\bR P^{3}$. \item Prove that a smooth closed $n$-manifold $M$ is $R$-orientable if and only its tangent bundle is $R$-orientable. \end{enumerate} \chapter{An introduction to $K$-theory} The first generalized cohomology theory to be discovered was $K$-theory, and it plays a vital role in the connection of algebraic topology to analysis and algebraic geometry. The fact that it is a generalized cohomology theory is a consequence of the Bott periodicity theorem, which is one of the most important and influential theorems in all of topology. We give some basic information about $K$-theory and, following Adams and Atiyah, we explain how the Adams operations in $K$-theory allow a quick solution to the ``Hopf invariant one problem.'' One implication is the purely algebraic theorem that the only possible dimensions of a real (not necessarily associative) division algebra are 1, 2, 4, and 8. We shall only discuss complex $K$-theory, although there is a precisely analogous construction of real $K$-theory $KO$. From the point of view of algebraic topology, real $K$-theory is a substantially more powerful invariant, but complex $K$-theory is usually more relevant to applications in other fields. \section{The definition of $K$-theory} Except where otherwise noted, we work with complex vector bundles throughout this chapter. Dimension will mean complex dimension and line bundles will mean complex line bundles. We consider the set $Vect(X)$\index{Vect(X)@$Vect(X)$} of equivalence classes of vector bundles over a space $X$. We assume unless otherwise specified that $X$ is compact. We remind the reader that vector bundles can have different dimension over different components of $X$. The set $Vect(X)$ forms an Abelian monoid (= semi-group) under Whitney sum, and it forms a semi-ring with multiplication given by the (internal) tensor product of vector bundles over $X$. There is a standard construction, called the Grothendieck construction,\index{Grothendieck construction} of an Abelian group $G(M)$ associated to an Abelian monoid $M$: one takes the quotient of the free Abelian group generated by the elements of $M$ by the subgroup generated by the set of elements of the form $m+n-m\oplus n$, where $\oplus$ is the sum in $M$. The evident morphism of Abelian monoids $i: M\rtarr G(M)$ is universal: for any homomorphism of monoids $f: M\rtarr G$, where $G$ is an Abelian group, there is a unique homomorphism of groups $\tilde{f}: G(M)\rtarr G$ such that $\tilde{f}\com i=f$. If $M$ is a semi-ring, then its multiplication induces a multiplication on $G(M)$ such that $G(M)$ is a ring, called the Grothendieck ring\index{Grothendieck ring} of $M$. If the semi-ring $M$ is commutative, then the ring $G(M)$ is commutative. \begin{defn} The $K$-theory\index{K-theory@$K$-theory} of $X$, denoted $K(X)$,\index{K(X)@$K(X)$} is the Grothendieck ring of the semi-ring $Vect(X)$. An element of $K(X)$ is called a virtual bundle\index{virtual bundle} over $X$. We write $[\xi]$ for the element of $K(X)$ determined by a vector bundle $\xi$. \end{defn} Since $\epz$ is the identity element for the product in $K(X)$, it is standard to write $q=[\epz^q]$, where $\epz^q$ is the $q$-dimensional trivial bundle. For vector bundles over a based space $X$, we have the function $d: Vect(X)\rtarr \bZ$ that sends a vector bundle to the dimension of its restriction to the component of the basepoint $*$. Since $d$ is a homomorphism of semi-rings, it induces a dimension function\index{dimension function} $d: K(X)\rtarr \bZ$, which is a homomorphism of rings. Since $d$ is an isomorphism when $X$ is a point, $d$ can be identified with the induced map $K(X)\rtarr K(*)$. \begin{defn} The reduced $K$-theory\index{K-theory@$K$-theory!reduced} $\tilde{K}(X)$\index{K(X)a@$\tilde K(X)$} of a based space $X$ is the kernel of $d: K(X)\rtarr \bZ$. It is an ideal of $K(X)$ and thus a ring without identity. Clearly $K(X)\iso \tilde{K}(X)\times \bZ$. \end{defn} We have a homotopical interpretation of these definitions, and it is for this that we need $X$ to be compact. By the classification theorem, we know that $\sE U_n(X)$ is naturally isomorphic to $[X_+,BU(n)]$; we have adjoined a disjoint basepoint because we are thinking cohomologically and want the brackets to denote based homotopy classes of maps. We have maps $i_n: BU(n)\rtarr BU(n+1)$. With our construction of classifying spaces via Grassmannians, these maps are inclusions, and we define $BU$ to be the colimit of the $BU(n)$, with the topology of the union. We say that bundles $\ze$ and $\xi$ are stably equivalent\index{stably equivalent bundles} if, for a sufficiently large $q$, the bundles $\ze\oplus \epz^{q-m}$ and $\xi\oplus \epz^{q-n}$ are equivalent, where $m=d(\ze)$ and $n=d(\xi)$. Let $\sE U(X)$\index{EU(X)@$\sE U(X)$} be the set of stable equivalence classes of vector bundles over $X$. If $X$ is connected, or if we restrict attention to vector bundles that are $n$-plane bundles for some $n$, then $\sE U$ is isomorphic to $\colim \sE U_n(X)$, where the colimit is taken over the maps $\sE U_n(X)\rtarr \sE U_{n+1}(X)$ obtained by sending a bundle $\xi$ to $\xi\oplus \epz$. Since a map from a compact space $X$ into $BU$ has image in one of the $BU(n)$, and similarly for homotopies, we see that in this case $[X_+,BU]\iso \colim [X_+,BU(n)]$ and therefore $$\sE U(X)\iso [X_+,BU].$$ A deeper use of compactness gives the following basic fact. \begin{prop} If $\xi:E\rtarr X$ is a vector bundle over $X$, then there is a bundle $\et$ over $X$ such that $\xi\oplus \et$ is equivalent to $\epz^q$ for some $q$. \end{prop} \begin{proof}[Sketch proof] The space $\GA E$ of sections of $E$ is a vector space under fiberwise addition and scalar multiplication. Using a partition of unity argument, one can show that there is a finite dimensional vector subspace $V$ of $\GA(E)$ such that the map $g: X\times V\rtarr E$ specified by $g(x,s)=s(x)$ is an epimorphism of bundles over $X$. The resulting short exact sequence of vector bundles, like any other short exact sequence of vector bundles, splits as a direct sum, and the conclusion follows. \end{proof} \begin{cor} Every virtual bundle over $X$ can be written in the form $[\xi] - q$ for some bundle $\xi$ and non-negative integer $q$. \end{cor} \begin{proof} Given a virtual bundle $[\om] -[\ze]$, where $\om$ and $\ze$ are bundles, choose $\et$ such that $\ze\oplus \et \iso \epz^q$ and let $\xi = \om\oplus \et$. Then $[\om] -[\ze] = [\xi] - q$ in $K(X)$. \end{proof} \begin{cor} There is a natural isomorphism $\sE U(X)\rtarr \tilde{K}(X)$. \end{cor} \begin{proof} Writing $\sset{\xi}$ for the stable equivalence class of a bundle $\xi$, the required isomorphism is given by the correspondence $\sset{\xi} \leftrightarrow [\xi] - d(\xi)$. \end{proof} \begin{cor} Give $\bZ$ the discrete topology. For compact spaces $X$, there is a natural isomorphism $$K(X)\iso [X_+,BU\times \bZ].$$ For nondegenerately based compact spaces $X$, there is a natural isomorphism $$ \tilde{K}(X)\iso [X,BU\times \bZ].$$ \end{cor} \begin{proof} When $X$ is connected, the first isomorphism sends $[\xi]-q$ to $(f,n-q)$, where $\xi$ is an $n$-plane bundle with classifying map $f: X \rtarr BU(n)\subset BU$. The isomorphism for non-connected spaces follows since both functors send disjoint unions to Cartesian products. The second isomorphism follows from the first since $d: K(X)\rtarr \bZ$ can be identified with the map $[X_+,BU\times \bZ]\rtarr [S^0,BU\times \bZ]$ induced by the cofibration $S^0\rtarr X_+$, and the latter has kernel $[X,BU\times \bZ]$ since $X_+/S^0=X$. \end{proof} For general, non-compact, spaces $X$, it is best to define $K$-theory to mean represented $K$-theory. Here we implicitly apply CW approximation, or else use the definition in the following form. \begin{defn}\index{K-theory@$K$-theory!represented} For a space $X$ of the homotopy type of a CW complex, define $$ K(X) = [X_+,BU\times \bZ].$$ For a nondegenerately based space of the homotopy type of a CW complex, define $$ \tilde{K}(X) = [X,BU\times \bZ].$$ \end{defn} When $X$ is compact, we know that $K(X)$ is a ring. It is natural to expect this to remain true for general $X$. That this is the case is a direct consequence of the following result, which the reader should regard as an aside. \begin{prop} The space $BU\times \bZ$ is a ring space\index{ring space} up to homotopy. That is, there are additive and multiplicative $H$-space structures on $BU\times \bZ$ such that the associativity, commutativity, and distributivity diagrams required of a ring commute up to homotopy. \end{prop} \begin{proof}[Indications of proof] By passage to colimits over $m$ and $n$, the maps $p_{m,n}: BU(m)\times BU(n) \rtarr BU(m+n)$ induce an ``addition'' $\oplus: BU\times BU\rtarr BU$. In fact, we can define $BU$ in terms of planes in any copy of $\bC^{\infty}$, and the explicit maps $p_{m,n}$ of Chapter 23 \S2 pass to colimits to give $$G_{\infty}(\bC^{\infty})\times G_{\infty}(\bC^{\infty}) \rtarr G_{\infty}(\bC^{\infty}\oplus\bC^{\infty});$$ use of an isomorphism $\bC^{\infty}\oplus\bC^{\infty}\iso \bC^{\infty}$ gives the required map $\oplus$, which is well defined, associative, and commutative up to homotopy; the zero-dimensional plane provides a convenient basepoint $0$ with which to check that we have a zero element up to homotopy. Using ordinary addition on $\bZ$, we obtain the additive $H$-space structure on $BU\times \bZ$. Tensor products of universal bundles give rise to classifying maps $q_{m,n}: BU(m)\times BU(n)\rtarr BU(mn)$. These do not pass to colimits so readily, since one must take into account the bilinearity of the tensor product, for example the relation $(\ga_m\oplus\epz)\ten \ga_n \iso (\ga_m\ten\ga_n)\oplus \ga_n$, and we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product on $BU\times \bZ$. It actually factors through the smash product with respect to the basepoint $0$, since that acts as zero for the tensor product, and it restricts to an $H$-space structure on $BO\times \sset{1}$ with basepoint $(0,1)$. \end{proof} The study of ring spaces such as this is a relatively new, and quite deep, part of algebraic topology. However, the reader should feel reasonably comfortable with the additive $H$-space structure on $BU$. \section{The Bott periodicity theorem} There are various ways to state, and various ways to prove, this basic result. We describe several versions and implications. One starting point is the following calculation. We have a canonical line bundle $\ga_1^2$ over $S^2\iso \bC P^1$; its points are pairs $(L,x)$, where $L$ is a line in $\bC^2$ and $x$ is a point on that line. We let $H=\Hom(\ga_1^2,\epz)$ denote its dual. \begin{thm} $K(S^2)$ is generated as a ring by $[H]$ subject to the single relation $([H]-1)^2=0$. Therefore, as Abelian groups, $K(S^2)$ is free on the basis $\sset{1,[H]}$ and $\tilde{K}(S^2)$ is free on the basis $\sset{1-[H]}$. \end{thm} \begin{proof}[Indication of proof] We think of $S^2$ as the one-point compactification of $\bC$ decomposed as the union of the unit disk $D$ and the complement $D'$ of the interior of $D$, so that $D\cap D'=S^1$. Any $n$-plane bundle over $S^2$ restricts to a trivial bundle over $D$ and $D'$, and these trivial bundles restrict to the same bundle over $S^1$. Conversely, an isomorphism $f$ from the trivial bundle over $S^1$ to itself gives a way to glue together the trivial bundles over $D$ and $D'$ to reconstruct a bundle over $S^2$. Say that two such ``clutching functions''\index{clutching function} $f$ are equivalent if the bundles they give rise to are equivalent. A careful analysis of the form of the possible clutching functions $f$ leads to a canonical example in each equivalence class and thus to the required calculation. \end{proof} For any pair of spaces $X$ and $Y$, we have a K\"unneth-type ring homomorphism\index{Kunneth map@K\"unneth map} $$\al: K(X)\ten K(Y)\rtarr K(X\times Y)$$ specified by $\al(x\ten y) = \pi_1^*(x)\pi_2^*(y)$. \begin{thm}[Bott periodicity]\index{Bott periodicity} For compact spaces $X$, $$\al: K(X)\ten K(S^2) \rtarr K(X\times S^2)$$ is an isomorphism. \end{thm} \begin{proof}[Indication of proof] The restrictions to $X\times D$ and $X\times D'$ of a bundle over $X\times S^2$ are equivalent to pullbacks of bundles over $X$, and their further restrictions to $S^1$ are equivalent. Conversely, bundles $\ze$ and $\xi$ over $X$ together with an equivalence $f$ between the restrictions to $X\times S^1$ of the pullbacks of $\ze$ and $\xi$ to $X\times D$ and $X\times D'$ determine a bundle over $X\times S^2$. Again, a careful analysis, which is similar to that in the special case when $X=pt$, of the equivalence classes of the possible clutching data $(\ze,f,\xi)$ leads to the conclusion. \end{proof} The following useful observation applies to any representable functor, not just $K$-theory. \begin{lem} For nondegenerately based spaces $X$ and $Y$, the projections of $X\times Y$ on $X$ and on $Y$ and the quotient map $X\times Y\rtarr X\sma Y$ induce a natural isomorphism $$\tilde K(X\sma Y)\oplus \tilde K(X) \oplus \tilde K(Y) \iso \tilde K(X\times Y),$$ and $\tilde K(X\sma Y)$ is the kernel of the map $\tilde K(X\times Y)\rtarr \tilde K(X)\oplus \tilde K(Y)$ induced by the inclusions of $X$ and $Y$ in $X\times Y$. \end{lem} \begin{proof} The inclusion $X\wed Y\rtarr X\times Y$ is a cofibration with quotient $X\sma Y$, and $X$ and $Y$ are retracts of $X\times Y$ via the inclusions and projections. \end{proof} It follows easily that the K\"unneth map\index{Kunneth map@K\"unneth map} $\al: K(X)\ten K(Y)\rtarr K(X\times Y)$ induces a reduced K\"unneth map $\be: \tilde K(X)\ten \tilde K(Y)\rtarr \tilde K(X\sma Y)$. We have a splitting $$ \tilde K(X)\ten \tilde K(Y) \oplus \tilde K(X) \oplus \tilde K(Y) \oplus \bZ \iso K(X)\ten K(Y)$$ that is compatible with the splitting of the lemma. Therefore the following reduced form of the Bott periodicity theorem is equivalent to the unreduced form that we have already stated. \begin{thm}[Bott periodicity]\index{Bott periodicity} For nondegenerately based compact spaces $X$, $$\be: \tilde K(X)\ten \tilde K(S^2) \rtarr \tilde K(X\sma S^2) = \tilde K(\SI^2 X)$$ is an isomorphism. \end{thm} Write $b=1-[H]\in\tilde K(S^2)$. Since $\tilde K(S^2)\iso \bZ$ with generator $b$, the theorem implies that multiplication by the ``Bott element'' $b$ specifies an isomorphism $$[X,BU\times \bZ]\iso \tilde K(X) \rtarr \tilde K(\SI^2 X)\iso [X,\OM^2(BU\times \bZ)]$$ for nondegenerately based compact spaces $X$. Here the addition in the source and target is derived from the natural additive $H$-space structure on $BU\times \bZ$ on the left and the displayed double loop space on the right. If we had this isomorphism for general non-compact spaces $X$, we could apply it with $X=BU\times \bZ$ and see that it is induced by a homotopy equivalence of $H$-spaces $$\be: BU\times \bZ \rtarr \OM^2(BU\times\bZ).$$ In fact, one can deduce such a homotopy equivalence from the Bott periodicity theorem as just stated, but there are more direct proofs. On the right, the double loop space obviously depends only on the basepoint component $BU=BU\times\sset{0}$. Since $\pi_2(BU)=\bZ$, a little argument with $H$-spaces shows that $\OM^2(BU\times \bZ)$ is equivalent as an $H$-space to $(\OM^2_0 BU)\times \bZ$, where $\OM^2_0 BU$ denotes the component of the basepoint in $\OM^2 BU$. Using the identity function on the factor $\bZ$, we see that what is needed is an equivalence of $H$-spaces $\be: BU\rtarr \OM^2_0 BU$. In fact, it is easily deduced from the form of Bott periodicity that, up to homotopy, $\be$ must be the adjoint of the composite $$\xymatrix{ \SI^2 BU = BU\sma S^2 \ar[r]^-{\id\sma b} & BU\sma BU \ar[r]^-{\ten} & BU.}$$ The infinite unitary group $U$ is defined to be the union of the unitary groups $U(n)$, where $U(n)$ is embedded in $U(n+1)$ as matrices with last row and column zero except for $1$ on the diagonal. Then $\OM BU$ is homotopy equivalent as an $H$-space to $U$. Since $\pi_1(U)=\bZ$ and the universal cover of $U$ is the infinite special unitary group $SU$, $\OM U$ is equivalent as an $H$-space to $(\OM SU)\times \bZ$. Therefore $\be$ may be viewed as a map $BU\rtarr \OM SU$. Bott's original proof of the Bott periodicity theorem used the Grassmannian model for $BU$ to write down an explicit map $\be$ in the required homotopy class and then used Morse theory to prove that $\be$ is a homotopy equivalence. Bott's map $\be$ can also be proved to be a homotopy equivalence using only basic algebraic topology. Since $BU$ and $\OM SU$ are simply connected spaces of the homotopy types of CW complexes, a relative version of the Hurewicz theorem called the Whitehead theorem\index{Whitehead theorem} shows that $\be$ will be a weak equivalence and therefore a homotopy equivalence if it induces an isomorphism on integral homology. Since $H^*(BU(n))=\bZ[c_1,\ldots\!,c_n]$, $H^*(BU)\iso \bZ[c_i|i\geq 1]$. The $H$-space structure on $BU$ is induced by the maps $p_{m,n}$, and we find that the map $\ps: H^*(BU)\rtarr H^*(BU\times BU)\iso H^*(BU)\ten H^*(BU)$ induced by the product is given by $\ps(c_k)=\sum_{i+j=k}c_i\ten c_j$. A purely algebraic dualization argument proves that, as a ring, $$H_*(BU)\iso \bZ[\ga_i|i\geq 1],$$ where $\ga_i$ is the image of a generator of $H_{2i}(\bC P^{\infty})$ under the map induced by the inclusion of $\bC P^{\infty}=BU(1)$ in $BU$. One can calculate $H_*(\OM SU)$ and see that it too is a polynomial algebra with an explicitly given generator in each even degree. A direct inspection of the map $\be$ shows that it carries generators to generators. In any case, it should now be clear that we have a periodic $\OM$-prespectrum and therefore a generalized cohomology theory represented by it. \begin{defn} The $K$-theory $\OM$-prespectrum $KU$\index{KU@$KU$} has spaces $KU_{2i} = BU\times \bZ$ and $KU_{2i+1}=U$ for all $i\geq 0$. The structure maps are given by the canonical homotopy equivalence $U\htp \OM BU = \OM(BU\times \bZ)$ and the Bott equivalence $BU\times \bZ\htp \OM U$. \end{defn} We have a resulting reduced cohomology theory\index{K-theory@$K$-theory!periodic} on based spaces such that $\tilde K^{2i}(X) = \tilde K(X)$ and $\tilde K^{2i+1}(X) = \tilde K(\SI X)$ for all integers $i$. This theory has products that are induced by tensor products of bundles over compact spaces and that are induced by suitable maps $\ph: KU_i\sma KU_j\rtarr KU_{i+j}$ in general, just as for the cup product in ordinary cohomology. It is standard to view this simply as a $\bZ_2$-graded theory with groups $\tilde K^0(X)$ and $\tilde K^1(X)$. \section{The splitting principle and the Thom isomorphism} Returning to our bundle theoretic construction of $K$-theory, with $X$ compact, we describe briefly some important generalizations of the Bott periodicity theorem. The reader should recall the Thom isomorphism theorem in ordinary cohomology from Chapter 23 \S5. We let $\xi: E\rtarr X$ be an $n$-plane bundle over $X$, fixed throughout this section. (We shall use the letters $E$ and $\xi$ more or less interchangeably.) Results for general vector bundles over non-connected spaces $X$ can be deduced by applying the results to follow to one component of $X$ at a time. \begin{defn} Let $E_0$ be the zero section of $E$. Define the projective bundle\index{projective bundle} $\pi: P(E)\rtarr X$ by letting the non-zero complex numbers act on $E-E_0$ by scalar multiplication on fibers and taking the orbit space under this action. Equivalently, the fiber $\pi^{-1}(x)\subset P(E)$ is the complex projective space of lines through the origin in the fiber $\xi^{-1}(x)\subset E$. Define the canonical line bundle $L(E)$ over $P(E)$ to be the subbundle of the pullback $\pi^*E$ of $\xi$ along $\pi$ whose points are the pairs consisting of a line in a fiber of $E$ and a point on that line. Let $Q(E)$ be the quotient bundle $\pi^*E/L(E)$ and let $H(E)$ denote the dual of $L(E)$. \end{defn} Observe that $P(\epz^2)=X\times \bC P^1$ is the trivial bundle over $X$ with fiber $\bC P^1\iso S^2$. The first version of Bott periodicity generalizes, with essentially the same proof by analysis of clutching data, to the following version. Regard $K(P(E))$ as a $K(X)$-algebra via $\pi^*: K(X)\rtarr K(P(E))$. \begin{thm}[Bott periodicity]\index{Bott periodicity} Let $L$ be a line bundle over $X$ and let $H=H(L\oplus \epz)$. Then the $K(X)$-algebra $K(P(L\oplus\epz))$ is generated by the single element $[H]$ subject to the single relation $([H]-1)([L][H]-1)=0$. \end{thm} There is a further generalization to arbitrary bundles $E$. To place it in context, we shall first explain a cohomological analogue that expresses a different approach to the Chern classes than the one that we sketched before. It will be based on a generalization to projective bundles of the calculation of $H^*(\bC P^n)$. The proofs of both results are intertwined with the proof of the following ``splitting principle,'' which allows the deduction of explicit formulas about general bundles from formulas about sums of line bundles. \begin{thm}[Splitting principle]\index{splitting principle} There is a compact space $F(E)$ and a map $p: F(E)\rtarr X$ such that $p^*E$ is a sum of line bundles over $F(E)$ and both $p^*: H^*(X;\bZ)\rtarr H^*(F(E);\bZ)$ and $p^*: K(X)\rtarr K(F(E))$ are monomorphisms. \end{thm} This is an easy inductive consequence of the following result, which we shall refer to as the ``splitting lemma.'' \begin{lem}[Splitting lemma]\index{splitting lemma} Both $\pi^*: H^*(X;\bZ)\rtarr H^*(P(E);\bZ)$ and $\pi^*: K(X)\rtarr K(P(E))$ are monomorphisms. \end{lem} \begin{proof}[Proof of the splitting principle] The pullback $\pi^*E$ splits as the sum $L(E)\oplus Q(E)$. (The splitting is canonically determined by a choice of a Hermitian metric on $E$.) Applying this construction to the bundle $Q(E)$ over $P(E)$, we obtain a map $\pi: P(Q(E))\rtarr P(E)$ with similar properties. We obtain the desired map $p: F(E)\rtarr X$ by so reapplying the projective bundle construction $n$ times. Explicitly, using a Hermitian metric on $E$, we find that the fiber $F(E)_x$ is the space of splittings of the fiber $E_x$ as a sum of $n$ lines, and the points of the bundle $p^*E$ are $n$-tuples of vectors in given lines. The splitting lemma implies the desired monomorphisms on cohomology and $K$-theory. \end{proof} \begin{thm} Let $x=c_1(L(E))\in H^2(P(E);\bZ)$. Then $H^*(P(E);\bZ)$ is the free $H^*(X;\bZ)$-module on the basis $\sset{1,x,\ldots\!,x^{n-1}}$, and the Chern classes\index{Chern classes} of $\xi$ are characterized by $c_0(\xi)=1$ and the formula $$\sum_{k=0}^n(-1)^kc_k(E)x^{n-k}=0.$$ \end{thm} \begin{proof}[Sketch proof] This is another case where the Serre spectral sequence shows that the bundle behaves cohomologically as if it were trivial and the K\"unneth theorem applied. This gives the structure of $H^*(P(E))$ as an $H^*(X)$-module. In particular, it implies the splitting lemma and thus the splitting principle in ordinary cohomology. It also implies that there must be some description of $x^n$ as a linear combination of the $x^k$ for $k 1$. The definition of $ch$ implies that the component $ch_n$ of $ch$ in degree $2n$ is $c_n/(n-1)!$ plus terms decomposable in terms of the $c_i$ for $i 1$. The exterior powers\index{exterior powers} of bundles satisfy the relation $$\la^k(\xi\oplus \et)= \oplus_{i+j=k}\la^i(\xi)\ten \la^j(\et).$$ It follows formally that the $\la^k$ extend to operations $K(X)\rtarr K(X)$. Indeed, form the group $G$ of power series with constant coefficient $1$ in the ring $K(X)[[t]]$ of formal power series in the variable $t$. We define a function from (equivalence classes of) vector bundles to this Abelian group by setting $$\LA(\xi)=1 + \la^1(\xi)t +\cdots + \la^k(\xi)t^k +\cdots.$$ Visibly, this is a morphism of monoids, $$\LA(\xi\oplus\et) = \LA(\xi)\LA(\et).$$ It therefore extend to a homomorphism of groups $\LA: K(X)\rtarr G$, and we let $\la^k(x)$ be the coefficient of $t^k$ in $\LA(x)$. We define the $\ps^k$ as suitable polynomials in the $\la^k$. Recall that the subring of symmetric polynomials in the polynomial algebra $\bZ[x_1,\ldots\!,x_n]$ is the polynomial algebra $\bZ[\si_1,\ldots\!,\si_n]$, where $\si_i=x_1x_2\cdots x_i+\cdots$ is the $i$th elementary symmetric function. We may write the power sum $\pi_k=x_1^k+\cdots+x_n^k$ as a polynomial $$\pi_k = Q_k(\si_1,\ldots\!,\si_k)$$ in the first $k$ elementary symmetric functions. Provided $n\geq k$, $Q_k$ does not depend on $n$. We define $$\ps^k(x) = Q_k(\la^1(x),\ldots\!, \la^k(x)).$$ For example, $\pi_2=\si_1^2-2\si_2$, hence $\ps^2(x)=x^2-2\la^2(x)$. The naturality of the $\ps^k$ is clear from the naturality of the $\la^k$. If $\xi$ is a line bundle, then $\la^1(\xi)=\xi$ and $\la^k(\xi)=0$ for $k\geq 2$. Clearly $\si^k_1 = \pi_k + \text{other terms}$ and $\pi_k$ does not occur as a summand of any other monomial in the $\si_i$. Therefore $Q_k \equiv \si_1^k$ modulo terms in the ideal generated by the $\si_i$ for $i>1$. This immediately implies property 4. Moreover, if $\xi_1,\ldots\!,\xi_n$ are line bundles, then \begin{eqnarray*} \LA (\xi_1\oplus \cdots \oplus \xi_n) & = & (1+\xi_1t)\cdots(1+\xi_nt) \\ & = & 1+\si_1(\xi_1,\ldots\!,\xi_n)t+\si_2(\xi_1,\ldots\!,\xi_n)t^2 + \cdots. \end{eqnarray*} This implies the generalization of property 4 to sums of line bundles: \begin{enumerate} \item[$4'$] $\ps^k(\xi_1\oplus\cdots\oplus \xi_n)= \pi_k(\xi_1,\ldots\!, \xi_n)$ for line bundles $\xi_i$. \end{enumerate} Now, if $x$ and $y$ are sums of line bundles, the following formulas are immediate: $$ \ps^k(x+y) = \ps^k(x)+\ps^k(y), \ \ \ps^k(xy) = \ps^k(x)\ps^k(y),\ \ \ps^k\ps^{\ell}(x)=\ps^{k\ell}(x) $$ $$\text{and}\ \ \ps^p(x)\equiv x^p\ \text{mod}\ p \ \ \text{for a prime}\ p. $$ For arbitrary bundles, these formulas follow directly from the splitting principle and naturality, and they then follow formally for arbitrary virtual bundles. This completes the proof of all properties except 5. We have that $\tilde{K}(S^2)$ is generated by $1-[H]$, where $(1-[H])^2=0$. Clearly $\ps^k(1-[H]) = 1-[H]^k$. By induction on $k$, $1-[H]^k=k(1-[H])$. Since $S^{2n}=S^2\sma\cdots\sma S^2$ and $\tilde{K}(S^{2n})$ is generated by the $k$-fold external tensor power $(1-[H])\ten\cdots\ten(1-[H])$, property 5 follows from the fact that $\ps^k$ preserves products. \begin{rem} By the splitting principle, it is clear that the $\ps^k$ are the unique natural and additive operations with the specified behavior on line bundles. \end{rem} Two further properties of the $\ps^k$ should be mentioned. The first is a direct consequence of the multiplicativity of the $\ps^k$ and their behavior on spheres. \begin{prop} The following diagram does not commute for based spaces $X$, where $\be$ is the periodicity isomorphism: $$\diagram \tilde K(X)\dto_{\ps^k} \rto^(0.43){\be} & \tilde K(\SI^2 X) \dto^{\ps^k}\\ \tilde K(X) \rto_(0.43){\be} & \tilde K(\SI^2 X).\\ \enddiagram$$ Rather, $\ps^k\be=k\be \ps^k$. \end{prop} Therefore the $\ps^k$ do not give stable operations on the $\bZ$-graded theory $K^*$. \begin{prop} Define $\ps^k_H$ on $H^{even}(X;\bZ)$ by letting $\ps^k_H(x) = k^rx$ for $x\in H^{2r}(X;\bZ)$. Then the following diagram commutes: $$\diagram K(X)\dto_{\ps^k} \rto^(0.35){ch} & H^{even}(X;\bQ)\dto^{\ps^k_H}\\ K(X) \rto_(0.35){ch} & H^{even}(X;\bQ).\\ \enddiagram$$ \end{prop} \begin{proof} It suffices to prove this on vector bundles $E$. By the splitting principle in $K$-theory and cohomology, we may assume that $E$ is a sum of line bundles. By additivity, we may then assume that $E$ is a line bundle. Here $\ps^k(E)= E^k$ and $c_1(E^k) = kc_1(E)$. The conclusion follows readily from the definition of $ch$ in terms of $e^t$. \end{proof} \begin{rem} The observant reader will have noticed that, by analogy with the definition of the Stiefel-Whitney classes, we can define characteristic classes\index{characteristic classes!in $K$-theory} in $K$-theory by use of the Adams operations and the Thom isomorphism, setting $\rh^k(E) = \PH^{-1}\ps^k\PH(1)$ for $n$-plane bundles $E$. \end{rem} \section{The Hopf invariant one problem and its applications} We give one of the most beautiful and impressive illustrations of the philosophy described in the first chapter. We define a numerical invariant, called the ``Hopf invariant,'' of maps $f: S^{2n-1}\rtarr S^n$ and show that it can only rarely take the value one. We then indicate several problems whose solution can be reduced to the question of when such maps $f$ take the value one. Adams' original solution to the Hopf invariant one problem used secondary cohomology operations in ordinary cohomology and was a critical starting point of modern algebraic topology. The later realization that a problem that required secondary operations in ordinary cohomology could be solved much more simply using primary operations in $K$-theory had a profound impact on the further development of the subject. Take cohomology with integer coefficients unless otherwise specified. \begin{defn} Let $X$ be the cofiber of a based map $f: S^{2n-1}\rtarr S^n$, where $n\geq 2$. Then $X$ is a CW complex with a single vertex, a single $n$-cell $i$, and a single $2n$-cell $j$. The differential in the cellular chain complex of $X$ is zero for obvious dimensional reasons, hence $\tilde H^*(X)$ is free Abelian on generators $x=[i]$ and $y=[j]$. Define an integer $h(f)$, the Hopf invariant\index{Hopf invariant} of $f$, by $x^2 = h(f) y$. We usually regard $h(f)$ as defined only up to sign (thus ignoring problems of orientations of cells). Note that $h(f)$ depends only on the homotopy class of $f$. \end{defn} If $n$ is odd, then $2x^2 = 0$ and thus $x^2=0$. We assume from now on that $n$ is even. Although not essential to the main point of this section, we record the following basic properties of the Hopf invariant. \begin{prop} The Hopf invariant enjoys the following properties. \begin{enumerate} \item If $g: S^{2n-1}\rtarr S^{2n-1}$ has degree $d$, then $h(f\com g) = dh(f)$. \item If $e: S^n\rtarr S^n$ has degree $d$, then $h(e\com f)=d^2h(f)$. \item The Hopf invariant defines a homomorphism $\pi_{2n-1}(S^n)\rtarr \bZ$. \item There is a map $f: S^{2n-1}\rtarr S^n$ such that $h(f)=2$. \end{enumerate} \end{prop} \begin{proof} We leave the first three statements to the reader. For property 4, let $\pi: D^n\rtarr D^n/S^{n-1}\iso S^n$ be the quotient map and define $$f: S^{2n-1}\iso (D^n\times S^{n-1})\cup (S^{n-1}\times D^n)\rtarr S^n$$ by $f(x,y)= \pi(x)$ and $f(y,x)=\pi(x)$ for $x\in D^n$ and $y\in S^{n-1}$. We leave it to the reader to verify that $h(f)=2$. \end{proof} We have adopted the standard definition of $h(f)$, but we could just as well have defined it in terms of $K$-theory. To see this, consider the cofiber sequence $$ S^{2n-1} \overto{f} S^n\overto{i} X \overto{\pi} S^{2n} \overto{\SI f} S^{n+1}.$$ Obviously $i^*: H^n(X)\rtarr H^n(S^n)$ and $\pi^*: H^{2n}(S^{2n})\rtarr H^{2n}(X)$ are isomorphisms. We have the commutative diagram with exact rows $$\diagram 0 \rto & \tilde K(S^{2n}) \dto_{ch} \rto^{\pi^*} & \tilde K(X) \dto^{ch} \rto^{i^*} & \tilde K(S^n) \dto^{ch}\rto & 0\\ 0 \rto & \tilde H^{**}(S^{2n};\bQ) \rto_{\pi^*} & \tilde H^{**}(X;\bQ) \rto_{i^*} & \tilde H^{**}(S^n;\bQ) \rto & 0.\\ \enddiagram$$ Here the top row is exact since $\tilde K^1(S^n)=0$ and $\tilde K^1(S^{2n})=0$. The vertical arrows are monomorphisms since they are rational isomorphisms. By a lemma in the previous section, generators $i_n$ of $\tilde K(S^n)$ and $i_{2n}$ of $\tilde K(S^{2n})$ map under $ch$ to generators of $H^n(S^n)$ and $H^{2n}(S^{2n})$. Choose $a\in \tilde K(X)$ such that $i^*(a) = i_n$ and let $b=\pi^*(i_{2n})$. Then $\tilde K(X)$ is the free Abelian group on the basis $\sset{a,b}$. Since $i_n^2=0$, we have $a^2=h'(f)b$ for some integer $h'(f)$. The diagram implies that, up to sign, $ch(b)= y$ and $ch(a) = x+ qy$ for some rational number $q$. Since $ch$ is a ring homomorphism and since $y^2=0$ and $xy=0$, we conclude that $h'(f)=h(f)$. \begin{thm} If $h(f)=\pm 1$, then $n = 2$, $4$, or $8$. \end{thm} \begin{proof} Write $n=2m$. Since $\ps^k(i_{2n})=k^{2m} i_{2n}$ and $\ps^k(i_n) = k^m i_n$, we have $$\ps^k(b)=k^{2m} b \ \ \tand \ \ \ps^k(a) = k^m a + \mu_k b$$ for some integer $\mu_k$. Since $\ps^2(a)\equiv a^2\ \text{mod}\ 2$, $h(f)=\pm 1$ implies that $\mu_2$ is odd. Now, for any odd $k$, \begin{eqnarray*} \ps^k\ps^2(a) & = & \ps^k(2^m a + \mu_2 b) \\ & = & k^m2^ma +(2^m\mu_k + k^{2m}\mu_2) b \end{eqnarray*} while \begin{eqnarray*} \ps^2\ps^k(a) & = & \ps^2(k^m a + \mu_k b) \\ & = & 2^mk^ma + (k^m\mu_2 + 2^{2m} \mu_k) b. \end{eqnarray*} Since these must be equal, we find upon equating the coefficients of $b$ that $$2^m(2^m-1)\mu_k = k^m(k^m-1)\mu_2.$$ If $\mu_2$ is odd, this implies that $2^m$ divides $k^m-1$. Already with $k=3$, an elementary number theoretic argument shows that this implies $m=1$, $2$, or $4$. \end{proof} This allows us to determine which spheres can admit an $H$-space structure. Recall from a problem in Chapter 18 that $S^{2m}$ cannot be an $H$-space. Clearly $S^n$ is an $H$-space for $n=0$, $1$, $3$, and $7$: view $S^n$ as the unit sphere in the space of real numbers, complex numbers, quaternions, or Cayley numbers. \begin{thm} If $S^{n-1}$ is an $H$-space,\index{Hspace@$H$-space} then $n=1$, $2$, $4$, or $8$. \end{thm} The strategy of proof is clear: given an $H$-space structure on $S^{n-1}$, we construct from it a map $f: S^{2n-1}\rtarr S^n$ of Hopf invariant one. The following construction and lemma do this and more. \begin{con}[Hopf construction]\index{Hopf construction} Let $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ be a map. Let $CX=(X\times I)/(X\times\sset{1})$ be the unreduced cone functor and note that we have canonical homeomorphisms of pairs $$(D^n,S^{n-1}) \iso(CS^{n-1},S^{n-1})$$ and \begin{eqnarray*} (D^{2n},S^{2n-1}) & \iso & (D^n\times D^n,(D^n\times S^{n-1})\cup (S^{n-1}\times D^n))\\ & \iso & (CS^{n-1}\times CS^{n-1},(CS^{n-1}\times S^{n-1})\cup (S^{n-1}\times CS^{n-1})). \end{eqnarray*} Take $S^n$ to be the unreduced suspension of $S^{n-1}$, with the upper and lower hemispheres $D^n_+$ and $D^n_-$ corresponding to the points with suspension coordinate $1/2\leq t\leq 1$ and $0\leq t\leq 1/2$, respectively. Define $$f: S^{2n-1}\iso (CS^{n-1}\times S^{n-1})\cup (S^{n-1}\times CS^{n-1}) \rtarr S^n$$ as follows. Let $x,y\in S^{n-1}$ and $t\in I$. On $CS^{n-1}\times S^{n-1}$, $f$ is the composite $$CS^{n-1}\times S^{n-1} \overto{\al} C(S^{n-1}\times S^{n-1}) \overto{C \ph} C S^{n-1} \overto{\be} D^n_-,$$ where $\al([x,t],y)=[(x,y),t]$ and $\be([x,t])=[x,(1-t)/2]$. On $S^{n-1}\times CS^{n-1}$, $f$ is the composite $$S^{n-1}\times CS^{n-1} \overto{\al'} C(S^{n-1}\times S^{n-1}) \overto{C \ph} C S^{n-1} \overto{\be'} D^n_+,$$ where $\al'(x,[y,t])=[(x,y),t]$ and $\be'([x,t])=[x,(1+t)/2]$. The map $f$, or rather the resulting $2$-cell complex $X=S^n\cup_f D^{2n}$, is called the Hopf construction on $\ph$. \end{con} Giving $S^{n-1}$ a basepoint, we obtain inclusions of $S^{n-1}$ onto the first and second copies of $S^{n-1}$ in $S^{n-1}\times S^{n-1}$. The bidegree\index{bidegree of a map} of a map $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ is the pair of integers given by the two resulting composite maps $S^{n-1}\rtarr S^{n-1}$. Thus $\ph$ gives $S^{n-1}$ an $H$-space structure if its bidegree is $(1,1)$. \begin{lem} If the bidegree of $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ is $(d_1,d_2)$, then the Hopf invariant of the Hopf construction on $\ph$ is $\pm d_1d_2$. \end{lem} \begin{proof} Making free use of the homeomorphisms of pairs specified in the construction, we see that the diagonal map of $X$, its top cell $j$, evident quotient maps, and projections $\pi_i$ onto first and second coordinates give rise to a commutative diagram in which the maps marked $\htp$ are homotopy equivalences and those marked $\iso$ are homeomorphisms: $$\diagram X \rto^{\DE} \dto & X\sma X \dto^{\htp} \\ X/S^n \rto^{\DE} & X/D^n_+ \sma X/D^n_- \\ S^{2n}\iso D^{2n}/S^{2n-1} \drto_{\iso} \rto^(0.3){\DE} \uto_j^{\iso} & (D^n\times D^n)/(S^{n-1}\times D^n) \sma (D^n\times D^n)/(D^n\times S^{n-1}) \uto_{j\sma j} \dto_{\htp}^{\pi_1\sma\pi_2} \\ & D^n/S^{n-1} \sma D^n/S^{n-1}\iso S^n\sma S^n. \\ \enddiagram$$ The cup square of $x\in H^n(X)$ is the image under $\DE^*$ of the external product of $x$ with itself. The maps on the left induce isomorphisms on $H^{2n}$. The inclusions of $D^n$ in the $i$th factor of $D^n\times D^n$ induce homotopy inverses $$\io_1: D^n/S^{n-1}\rtarr (D^n\times D^n)/(S^{n-1}\times D^n)$$ and $$\io_2: D^n/S^{n-1} \rtarr (D^n\times D^n)/(D^n\times S^{n-1})$$ to the projections $\pi_i$ in the diagram, and it suffices to prove that, up to sign, the composites $$j\com \io_1: D^n/S^{n-1}\rtarr X/D^n_+ \tand j\com\io_2: D^n/S^{n-1}\rtarr X/D^n_-$$ induce multiplication by $d_1$ and by $d_2$ on $H^n$. However, by construction, these maps factor as composites $$D^n/S^{n-1}\overto{\ga_1}S^n/D^n_+\rtarr X/D^n_+ \tand D^n/S^{n-1}\overto{\ga_2}S^n/D^n_-\rtarr X/D^n_-,$$ where, up to signs and identifications of spheres, $\ga_1$ and $\ga_2$ are the suspensions of the restrictions of $\ph$ to the two copies of $S^{n-1}$ in $S^{n-1}\times S^{n-1}$. \end{proof} The determination of which spheres are $H$-spaces has the following implications. \begin{thm}\index{products on $\bR^n$} Let $\om: \bR^n\times \bR^n\rtarr \bR^n$ be a map with a two-sided identity element $e\neq 0$ and no zero divisors. Then $n=1$, $2$, $4$, or $8$. \end{thm} \begin{proof} The product restricts to give $\bR^n-\sset{0}$ an $H$-space structure. Since $S^{n-1}$ is homotopy equivalent to $\bR^n-\sset{0}$, it inherits an $H$-space structure. Explicitly, we may assume that $e\in S^{n-1}$, by rescaling the metric, and we give $S^{n-1}$ the product $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ specified by $\ph(x,y)=\om(x,y)/|\om(x,y)|$. \end{proof} Note that $\om$ need not be bilinear, just continuous. Also, it need not have a strict unit; all that is required is that $e$ be a two-sided unit up to homotopy for the restriction of $\om$ to $\bR^n-\sset{0}$. \begin{thm} If $S^{n}$ is parallelizable,\index{parallelizable spheres} then $n=0$, $1$, $3$, or $7$. \end{thm} \begin{proof} Exclude the trivial case $n=0$ and suppose that $S^{n}$ is parallelizable, so that its tangent bundle $\ta$ is trivial. We will show that $S^{n}$ is an $H$-space. Define a map $\mu: \ta \rtarr S^{n}$ as follows. Think of the tangent plane $\ta_x$ as affinely embedded in $\bR^{n+1}$ with origin at $x$. We have a parallel translate of this plane to an affine plane with origin at $-x$. Define $\mu$ by sending a tangent vector $y\in \ta_x$ to the intersection with $S^{n}$ of the line from $x$ to the translate of $y$. Composing with a trivialization $S^{n}\times\bR^n\iso \ta$, this gives a map $\mu: S^{n}\times\bR^n\rtarr S^n$. Let $S^n_{\infty}$ be the one-point compactification of $\bR^n$. Extend $\mu$ to a map $\ph: S^n\times S^n_{\infty}\rtarr S^n$ by letting $\ph(x,\infty)=x$; $\ph$ is continuous since $\mu(x,y)$ approaches $x$ as $y$ approaches $\infty$. By construction, $\infty$ is a right unit for this product. For a fixed $x$, $y\rtarr \ph(x,y)$ is a degree one homeomorphism $S^n_{\infty}\rtarr S^n_{\infty}$. The conclusion follows. \end{proof} \chapter{An introduction to cobordism} Cobordism theories were introduced shortly after $K$-theory, and their use pervades modern algebraic topology. We shall describe the cobordism of smooth closed manifolds, but this is in fact a particularly elementary example. Other examples include smooth closed manifolds with extra structure on their stable normal bundles: orientation, complex structure, Spin structure, or symplectic structure for example. All of these except the symplectic case have been computed completely. The complex case is particularly important since complex cobordism and theories constructed from it have been of central importance in algebraic topology for the last few decades, quite apart from their geometric origins in the classification of manifolds. The area is pervaded by insights from algebraic topology that are quite mysterious geometrically. For example, the complex cobordism groups turn out to be concentrated in even degrees: every smooth closed manifold of odd dimension with a complex structure on its stable normal bundle is the boundary of a compact manifold (with compatible bundle information). However, there is no geometric understanding of why this should be the case. The analogue with ``complex'' replaced by ``symplectic'' is false. \section{The cobordism groups of smooth closed manifolds} We consider the problem of classifying smooth closed $n$-manifolds $M$. One's first thought is to try to classify them up to diffeomorphism, but that problem is in principle unsolvable. Thom's discovery that one can classify such manifolds up to the weaker equivalence relation of ``cobordism''\index{cobordism} is one of the most beautiful advances of twentieth century mathematics. We say that two smooth closed $n$-manifolds $M$ and $N$ are cobordant\index{cobordant manifolds} if there is a smooth compact manifold $W$ whose boundary is the disjoint union of $M$ and $N$, $\pa W = M\amalg N$. We write $\sN_n$\index{Naa@$\sN_n$} for the set of cobordism classes of smooth closed $n$-manifolds. It is convenient to allow the empty set $\emptyset$ as an $n$-manifold for every $n$. Disjoint union gives an addition on the set $\sN_n$. This operation is clearly associative and commutative and it has $\emptyset$ as a zero element. Since $$\pa(M\times I) =M\amalg M,$$ $M\amalg M$ is cobordant to $\emptyset$. Thus $M=-M$ and $\sN_n$ is a vector space over $\bZ_2$. Cartesian product of manifolds defines a multiplication $\sN_m\times \sN_n\rtarr \sN_{m+n}$. This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a single point provides an identity element. We conclude that $\sN_*$ is a graded $\bZ_2$-algebra. \begin{thm}[Thom]\index{Thom cobordism theorem} $\sN_*$ is a polynomial algebra over $\bZ_2$ on generators $u_i$ of dimension $i$ for $i > 1$ and not of the form $2^r-1$. \end{thm} As already stated in our discussion of Stiefel-Whitney numbers, it follows from the proof of the theorem that a manifold is a boundary if and only if its normal Stiefel-Whitney numbers are zero. We can restate this as follows.\index{Stiefel-Whitney numbers} \index{Stiefel-Whitney numbers!tangential}\index{Stiefel-Whitney numbers!normal} \begin{thm} Two smooth closed $n$-manifolds are cobordant if and only if their normal Stiefel-Whitney numbers, or equivalently their tangential Stiefel-Whitney numbers, are equal. \end{thm} Explicit generators $u_i$ are known. Write $[M]$ for the cobordism class of a manifold $M$. Then we can take $u_{2i}=[\bR P^{2i}]$. We have seen that the Stiefel-Whitney numbers of $\bR P^{2i-1}$ are zero, so we need different generators in odd dimensions. For $m q$, where $\ga_q^r$ is the restriction of the universal bundle $\ga_q$ to the compact manifold $G_q(\bR^r)$. By an implication of Sard's theorem known as the transversality\index{transversality} theorem, we can deform the restriction of $g$ to $g^{-1}(T\ga_q^r-\sset{\infty})=g^{-1}(E(\ga_q^r))$ so as to obtain a homotopic map that is smooth and transverse to the zero section. This use of transversality is the crux of the proof of the theorem. It follows that the inverse image $g^{-1}(G_q(\bR^r))$ is a smooth closed $n$-manifold embedded in $\bR^{n+q}=S^{n+q}-\sset{\infty}$. It is intuitively plausible that homotopic maps $g_i: S^{n+q}\rtarr TO(q)$, $i=0,\, 1$, give rise to cobordant $n$-manifolds by this construction. Indeed, with the $g_i$ smooth and transverse to the zero section, we can approximate a homotopy between them by a homotopy $h$ which is smooth on $h^{-1}(T(\ga_q^r)-\sset{\infty})$ and transverse to the zero section. Then $h^{-1}(G_q(\bR^{r}))$ is a manifold whose boundary is $g_0^{-1}(G_q(\bR^r))\amalg g_1^{-1}(G_q(\bR^r))$. It is easy to verify that the resulting function $\be: \pi_{n+q}(TO(q))\rtarr \sN_n$ is a homomorphism. If we start with a manifold $M$ embedded in $\bR^{n+q}$ and construct the classifying map $f$ for its normal bundle to be the Gauss map described in our sketch proof of the classification theorem in Chapter 23 \S1, then the composite $Tf\com t$ is smooth and transverse to the zero section, and the inverse image of the zero section is exactly $M$. This proves that $\be$ is an epimorphism. To complete the proof, it suffices to show that $\be$ is a monomorphism. It will follow formally that $\al$ is well defined and inverse to $\be$. Thus suppose given $g: S^{n+q}\rtarr T\ga_q^r$ such that $g^{-1}(E(\ga_q^r))$ is smooth and transverse to the zero section and suppose that $M=g^{-1}(G_q(\bR^r))$ is a boundary, say $M=\pa W$. The inclusion of $M$ in $S^{n+q}$ extends to a embedding of $W$ in $D^{n+q+1}$, by the Whitney embedding theorem for manifolds with boundary (assuming as always that $q$ is sufficiently large). We may assume that $U=g^{-1}(T\ga_q^r-\sset{\infty})$ is a tubular neighborhood and that $g: U \rtarr E(\ga_q^r)$ is a map of vector bundles. A relative version of the tubular neighborhood theorem then shows that $U$ can be extended to a tubular neighborhood $V$ of $W$ in $D^{n+q+1}$ and that $g$ extends to a map of vector bundles $h: V\rtarr E(\ga_q^r)$. We can then extend $h$ to a map $D^{n+q+1}\rtarr T(\ga_q^r)$ by mapping $D^{n+q+1}-V$ to $\infty$. This extension of $g$ to the disk implies that $g$ is null homotopic. We must still define the ring structure on $\pi_*(TO)$ and prove that we have an isomorphism of rings and therefore of $\bZ_2$-algebras. Recall that we have maps $p_{m,n}: BO(m)\times BO(n)\rtarr BO(m+n)$ such that $p_{m,n}^*(\ga_{m+n})=\ga_m\times \ga_n$. The Thom space $T(\ga_m\times\ga_n)$ is canonically homeomorphic to the smash product $TO(m)\sma TO(n)$, and the bundle map $\ga_m\times \ga_n \rtarr \ga_{m+n}$ induces a map $\ph_{m,n}: TO(m)\sma TO(n)\rtarr TO(m+n)$. If we have maps $f: S^{m+q}\rtarr TO(m)$ and $g: S^{n+q}\rtarr TO(n)$, then we can compose their smash product with $\ph_{m,n}$ to obtain a composite map $$ S^{m+n+q+r}\iso S^{m+q}\sma S^{n+r} \overto{f\sma g} TO(m)\sma TO(n) \overto{\ph_{m,n}} TO(m+n).$$ We can relate the maps $\ph_{m,n}$ to the maps $\si_n$. In fact, $TO$ is a commutative and associative ring prespectrum in the sense of the following definition. \begin{defn} Let $T$ be a prespectrum. Then $T$ is a ring prespectrum\index{ring prespectrum} \index{prespectrum!ring} if there are maps $\et: S^0 \rtarr T_0$ and $\ph_{m,n}: T_m\sma T_n\rtarr T_{m+n}$ such that the following diagrams are homotopy commutative: $$\diagram T_m\sma\SI T_n \ddouble \rto^{\id\sma\si_n} & T_m\sma T_{n+1} \drto^{\ph_{m,n+1}} & \\ \SI(T_m\sma T_n) \dto_{(-1)^n} \rto^(0.56){\SI\ph_{m,n}}& \SI T_{m+n} \rto^(0.44){\si_{m+n}} & T_{m+n+1} \\ (\SI T_m)\sma T_n \rto_(0.5){\si_m\sma \id} & T_{m+1}\sma T_n \urto_{\ph_{m+1,n}} & \\ \enddiagram$$ \vspace{.1in} $$\diagram S^0\sma T_n \rto^{\et\sma\id} \drto_{\iso} & T_0\sma T_n \dto^{\ph_{0,n}} \\ & T_n\\ \enddiagram \ \ \ \ \text{and} \ \ \ \ \diagram T_n\sma T_0 \dto_{\ph_{n,0}} & T_n\sma S^0; \lto_{\id\sma \et} \dlto^{\iso}\\ T_{n} & \\ \enddiagram$$ $T$ is associative if the following diagrams are homotopy commutative: $$\diagram T_m\sma T_n\sma T_p \dto_{\id\sma\ph_{n,p}}\rto^(0.53){\ph_{m,n}\sma\id} & T_{m+n}\sma T_p \dto^{\ph_{m+n,p}} \\ T_m\sma T_{n+p} \rto_{\ph_{m,n+p}} & T_{m+n+p}; \\ \enddiagram$$ $T$ is commutative if there are equivalences $(-1)^{mn}: T_{m+n}\rtarr T_{m+n}$ that suspend to $(-1)^{mn}$ on $\SI T_{m+n}$ and if the following diagrams are homotopy commutative: $$\diagram T_m\sma T_n \dto_{\ph_{m,n}}\rto^{t} & T_n\sma T_m \dto^{\ph_{n,m}} \\ T_{m+n} \rto_{(-1)^{mn}} & T_{m+n}.\\ \enddiagram$$ When $T$ is an $\OM$-prespectrum, we can restate this as $\ph_{m,n}\htp (-1)^{mn} \ph_{n,m}t$. \end{defn} For example, the Eilenberg-Mac\,Lane $\OM$-prespectrum of a commutative ring $R$ is an associative and commutative ring prespectrum by the arguments in Chapter 22 \S3. It is denoted $HR$ or sometimes, by abuse, $K(R,0)$. Similarly, the $K$-theory $\OM$-prespectrum is an associative and commutative ring prespectrum. The sphere prespectrum, whose $n$th space is $S^n$, is another example. For $TO$, the required maps $(-1)^{mn}: TO(m+n)\rtarr TO(m+n)$ are obtained by passage to Thom complexes from a map $\ga_{m+n}\rtarr \ga_{m+n}$ of universal bundles given on the domains of coordinate charts by the evident interchange isomorphism $\bR^{m+n}\rtarr \bR^{m+n}$. The following lemma is immediate by passage to colimits. \begin{lem} If $T$ is an associative ring prespectrum, then $\pi_*(T)$ is a graded ring. If $T$ is commutative, then $\pi_*(T)$ is commutative in the graded sense. \end{lem} Returning to the case at hand, we show that the maps $\al$ for varying $n$ transport products of manifolds to products in $\pi_*(TO)$. Thus let $M$ be an $m$-manifold embedded in $\bR^{m+q}$ with tubular neighborhood $U\iso E(\nu_M)$ and $N$ be an $n$-manifold embedded in $\bR^{n+r}$ with tubular neighborhood $V\iso E(\nu_N)$. Then $M\times N$ is embedded in $\bR^{m+q+n+r}$ with tubular neighborhood $U\times V\iso E(\nu_{M\times N})$. Identifying $S^{m+q+n+r}$ with $S^{m+q}\sma S^{n+r}$, we find that the Pontryagin-Thom construction for $M\times N$ is the smash product of the Pontryagin-Thom constructions for $M$ and $N$. That is, the left square in the following diagram commutes. The right square commutes up to homotopy by the definition of $\ph_{q,r}$. $$\diagram S^{m+q}\sma S^{n+r} \rto^{t\sma t} \ddouble & T\nu_m\sma T\nu_N \dto^{\iso} \rto & TO(q)\sma TO(r) \dto^{\ph_{q,r}} \\ S^{m+q+n+r} \rto_{t} & T(\nu_{M\times N}) \rto & TO(q+r).\\ \enddiagram$$ This implies the claimed multiplicativity of the maps $\al$. \section{Prespectra and the algebra $H_*(TO;\bZ_2)$} Calculation of the homotopy groups $\pi_*(TO)$ proceeds by first computing the homology groups $H_*(TO;\bZ_2)$ and then showing that the stable Hurewicz homomorphism maps $\pi_*(TO)$ monomorphically onto an identifiable part of $H_*(TO;\bZ_2)$. We explain the calculation of homology groups in this section and the next, connect the calculation with Stiefel-Whitney numbers in \S5, and describe how to complete the desired calculation of homotopy groups in \S6. We must first define the homology groups of prespectra and the stable Hurewicz homomorphism. Just as we defined the homotopy groups of a prespectrum $T$ by the formula $$\pi_n(T)=\colim \pi_{n+q}(T_q),$$ we define the homology and cohomology groups\index{prespectrum!homology groups of} \index{prespectrum!cohomology groups of} of $T$ with respect to a homology theory $k_*$ and cohomology theory $k^*$ on spaces by the formulas $$k_n(T)=\colim \tilde{k}_{n+q}(T_q),$$ where the colimit is taken over the maps $$ \tilde{k}_{n+q}(T_q) \overto{\SI_*} \tilde{k}_{n+q+1}(\SI T_q) \overto{{\si_q}_*} \tilde{k}_{n+q+1}(T_{q+1}),$$ and $$k^n(T) = \lim \tilde{k}^{n+q}(T_q),$$ where the limit is taken over the maps $$\tilde{k}^{n+q+1}(T_{q+1})\overto{\si^*_q} \tilde{k}^{n+q+1}(\SI T_q) \overto{\SI^{-1}} \tilde{k}^{n+q}(T_q).$$ In fact, this definition of cohomology is inappropriate in general, differing from the appropriate definition by a ${\lim}^1$ error term. However, the definition is correct when $k^*$ is ordinary cohomology with coefficients in a field $R$ and each $\tilde{H}^{n+q}(T_q;R)$ is a finite dimensional vector space over $R$. This is the only case that we will need in the work of this chapter. In this case, it is clear that $H^n(T;R)$ is the vector space dual of $H_n(T;R)$, a fact that we shall use repeatedly. Observe that there is no cup product in $H^*(T;R)$: the maps in the limit system factor through the reduced cohomologies of suspensions, in which cup products are identically zero (see Problem 5 at the end of Chapter 19). However, if $T$ is an associative and commutative ring prespectrum, then the homology groups $H_*(T;R)$ form a graded commutative $R$-algebra. The Hurewicz homomorphisms $\pi_{n+q}(T_q)\rtarr \tilde{H}_{n+q}(T_q;Z)$ pass to colimits to give the stable Hurewicz homomorphism\index{Hurewicz homomorphism!stable} $$h: \pi_n(T)\rtarr H_n(T;\bZ).$$ We may compose this with the map $H_n(T;\bZ)\rtarr H_n(T;R)$ induced by the unit of a ring $R$, and we continue to denote the composite by $h$. If $T$ is an associative and commutative ring prespectrum, then $h: \pi_*(T)\rtarr H_*(T;R)$ is a map of graded commutative rings. We shall write $H_*$ and $H^*$ for homology and cohomology with coefficients in $\bZ_2$ throughout \S\S3--6, and we tacitly assume that all homology and cohomology groups in sight are finite dimensional $\bZ_2$-vector spaces. Recall that we have Thom isomorphisms $$\PH_q: H^n(BO(q))\rtarr \tilde{H}^{n+q}(TO(q))$$ obtained by cupping with the Thom class $\mu_q\in \tilde{H}^q(TO(q))$. Naturality of the Thom diagonal applied to the map of bundles $\ga_q\oplus\epz \rtarr \ga_{q+1}$ gives the commutative diagram $$\diagram \SI TO(q) \rto^(0.4){\DE} \dto_{\si_q} & BO(q)_+\sma \SI TO(q) \dto^{i_q\sma \si_q}\\ TO(q+1) \rto_(0.35){\DE} & BO(q+1)_+\sma TO(q+1).\\ \enddiagram$$ This implies that the following diagram is commutative: $$\diagram H^n(BO(q+1))\rrto^{i_q^*} \dto_{\PH_{q+1}} & & H^n(BO(q)) \dto^{\PH_q}\\ \tilde{H}^{n+q+1}(TO(q+1))\rto_{\si^*_q} & \tilde{H}^{n+q+1}(\SI TO(q)) \rto_{\SI^{-1}} & \tilde{H}^{n+q}(TO(q)).\\ \enddiagram$$ We therefore obtain a ``stable Thom isomorphism''\index{Thom isomorphism!stable} $$\PH: H^n(BO)\rtarr H^n(TO)$$ on passage to limits. We have dual homology Thom isomorphisms $$\PH_n: \tilde{H}_{n+q}(TO(q))\rtarr H_n(BO(q))$$ that pass to colimits to give a stable Thom isomorphism $$\PH: H_n(T) \rtarr H_n(BO).$$ Naturality of the Thom diagonal applied to the map of bundles $\ga_q\oplus\ga_r \rtarr \ga_{q+r}$ gives the commutative diagram $$\diagram TO(q)\sma TO(r) \ddto_{\ph_{q,r}} \rto^(0.33){\DE\sma\DE} & BO(q)_+\sma TO(q)\sma BO(r)_+ \sma TO(r) \dto^{\id\sma t\sma \id}\\ & (BO(q)\times BO(r))_+\sma TO(q)\sma TO(r) \dto^{(p_{q,r})_+\sma \ph_{q,r}} \\ TO(q+r) \rto_(0.38){\DE} & BO(q+r)_+\sma TO(q+r). \\ \enddiagram$$ As we observed for $BU$ in the previous chapter, the maps $p_{q,r}$ pass to colimits to give $BO$ an $H$-space structure, and it follows that $H_*(BO)$ is a $\bZ_2$-algebra. On passage to homology and colimits, these diagrams imply the following conclusion. \begin{prop} The Thom isomorphism $\PH: H_*(TO)\rtarr H_*(BO)$ is an isomorphism of $\bZ_2$-algebras. \end{prop} The description of the $H^*(BO(n))$ and the maps $i_q^*$ in Chapter 23 \S2 implies that $$H^*(BO)=\bZ_2[w_i|i\geq 1]$$ as an algebra. However, we are more interested in its ``coalgebra''\index{coalgebra} structure, which is given by the vector space dual $$\ps: H^*(BO)\rtarr H^*(BO)\ten H^*(BO)$$ of its product in homology. It is clear from the description of the $p_{q,r}^*$ that $$\ps(w_k)=\sum_{i+j=k} w_i\ten w_j.$$ From here, determination of $H_*(BO)$ and therefore $H_*(TO)$ as an algebra is a purely algebraic, but non-trivial, problem in dualization. Let $i: \bR P^{\infty}=BO(1)\rtarr BO$ be the inclusion. Let $x_i\in H_i(\bR P^{\infty})$ be the unique non-zero element and let $b_i=i_*(x_i)$. Then the solution of our dualization problem takes the following form. \begin{thm} $H_*(BO)$ is the polynomial algebra $\bZ_2[b_i|i\geq 1]$. \end{thm} Let $a_i\in H_i(TO)$ be the element characterized by $\PH(a_i) = b_i$. \begin{cor} $H_*(TO)$ is the polynomial algebra $\bZ_2[a_i|i\geq 1]$. \end{cor} Using the compatibility of the Thom isomorphisms for $BO(1)$ and $BO$, we see that the $a_i$ come from $H_*(TO(1))$. Remember that elements of $H_{i+1}(TO(1))$ map to elements of $H_i(TO)$ in the colimit; in particular, the non-zero element of $H_1(TO(1))$ maps to the identity element $1\in H_0(TO)$. Recall from Chapter 23 \S6 that we have a homotopy equivalence $j: \bR P^{\infty}\rtarr TO(1)$. \begin{cor} For $i\geq 0$, $j_*(x_{i+1})$ maps to $a_i$ in $H_*(TO)$, where $a_0=1$. \end{cor} \section{The Steenrod algebra and its coaction on $H_*(TO)$} Since the Steenrod operations are stable and natural, they pass to limits to define natural operations\index{prespectrum!Steenrod operations of} $Sq^i: H^n(T)\rtarr H^{n+i}(T)$ for $i\geq 0$ and prespectra $T$. Here $Sq^0=\id$, but it is not true that $Sq^i(x)=0$ for $i>\deg\,x$. For example, we have the ``stable Thom class''\index{Thom class!stable} $\PH(1)=\mu\in H^0(TO)$, and it is immediate from the definition of the Stiefel-Whitney classes that $\PH(w_i)=Sq^i(\mu)$. Of course, $Sq^i(1)=0$ for $i>0$, so that $\PH$ does not commute with Steenrod operations. The homology and cohomology of $TO$ are built up from $\pi_*(TO)$ and Steenrod operations. We need to make this statement algebraically precise to determine $\pi_*(TO)$, and we need to assemble the Steenrod operations into an algebra to do this. \begin{defn} The mod $2$ Steenrod algebra\index{Steenrod algebra} $A$ is the quotient of the free associative $\bZ_2$-algebra generated by elements $Sq^i$, $i\geq 1$, by the ideal generated by the Adem relations (which are stated in Chapter 22 \S5). \end{defn} The following lemmas should be clear. \begin{lem} For spaces $X$, $H^*(X)$ has a natural $A$-module structure. \end{lem} \begin{lem} For prespectra $T$, $H^*(T)$ has a natural $A$-module structure. \end{lem} The elements of $A$ are stable mod $2$ cohomology operations, and our description of the cohomology of $K(\bZ_2,q)$s in Chapter 22 \S5 implies that $A$ is in fact the algebra of all stable mod $2$ cohomology operations, with multiplication given by composition. Passage to limits over $q$ leads to the following lemma. Alternatively, with the more formal general definitions of the next section, it will become yet another application of the Yoneda lemma. Recall that $H\bZ_2$ denotes the Eilenberg-Mac\,Lane $\OM$-prespectrum $\sset{K(\bZ_2,q)}$. \begin{lem} As a vector space, $A$ is isomorphic to $H^*(H\bZ_2)$. \end{lem} We shall see how to describe the composition in $A$ homotopically in the next section. What is more important at the moment is that the lemma allows us to read off a basis for $A$. \begin{thm} $A$ has a basis consisting of the operations $Sq^I = Sq^{i_1}\cdots Sq^{i_j}$, where $I$ runs over the sequences $\sset{i_1,\ldots\!,i_j}$ of positive integers such that $i_{r}\geq 2 i_{r+1}$ for $1\leq r < j$. \end{thm} What is still more important to us is that $A$ not only has the composition product $A\ten A\rtarr A$, it also has a coproduct $\ps: A\rtarr A\ten A$. Giving $A\ten A$ its natural structure as an algebra, $\ps$ is the unique map of algebras specified on generators by $\ps(Sq^k) = \sum_{i+j=k} Sq^i\ten Sq^j$. The fact that $\ps$ is a well defined map of algebras is a formal consequence of the Cartan formula. Algebraic structures like this, with compatible products and coproducts, are called ``Hopf algebras.''\index{Hopf algebra} We write $A_*$ for the vector space dual of $A$, and we give it the dual basis to the basis just specified on $A$. While $A_*$ is again a Hopf algebra, we are only interested in its algebra structure at the moment. In contrast with $A$, the algebra $A_*$ is commutative, as is apparent from the form of the coproduct on the generators of $A$. Recall that $H\bZ_2$ is an associative and commutative ring prespectrum, so that $H_*(H\bZ_2)$ is a commutative $\bZ_2$-algebra. The definition of the product on $H\bZ_2$ (in Chapter 22 \S3) and the Cartan formula directly imply the following observation. \begin{lem} $A_*$ is isomorphic as an algebra to $H_*(H\bZ_2)$. \end{lem} We need an explicit description of this algebra. In principle, this is a matter of pure algebra from the results already stated, but the algebraic work is non-trivial. \begin{thm} For $r\geq 1$, define $I_r=(2^{r-1}, 2^{r-2},\ldots\!, 2, 1)$ and define $\xi_r$ to be the basis element of $A_*$ dual to $Sq^{I_r}$. Then $A_*$ is the polynomial algebra $\bZ_2[\xi_r|r\geq 1]$. \end{thm} We need a bit of space level motivation for the particular relevance of the elements $\xi_r$. We left the computation of the Steenrod operations in $H^*(\bR P^{\infty})$ as an exercise, and the reader should follow up by proving the following result. \begin{lem} In $H^*(\bR P^{\infty})=\bZ_2[\al]$, $Sq^{I_r}(\al)=\al^{2^r}$ for $r\geq 1$ and $Sq^{I}(\al)=0$ for all other basis elements $Sq^I$ of $A$. \end{lem} The $A$-module structure maps $$A\ten H^*(X)\rtarr H^*(X) \ \ \tand \ \ A\ten H^*(T)\rtarr H^*(T)$$ for spaces $X$ and prespectra $T$ dualize to give ``$A_*$-comodule''\index{comodule} structure maps $$\ga: H_*(X)\rtarr A_*\ten H_*(X) \ \ \tand \ \ \ga: H_*(T)\rtarr A_*\ten H_*(T).$$ We remind the reader that we are implicitly assuming that all homology and cohomology groups in sight are finitely generated $\bZ_2$-vector spaces, although these ``coactions'' can in fact be defined without this assumption. Formally, the notion of a comodule $N$ over a coalgebra $C$ is defined by reversing the direction of arrows in a diagrammatic definition of a module over an algebra. For example, for any vector space $V$, $C\ten V$ is a comodule with action $$\ps\ten\id: C\ten V\rtarr C\ten C\ten V.$$ Note that, dualizing the unit of an algebra, a $\bZ_2$-coalgebra is required to have a counit $\epz: C\rtarr \bZ_2$. We understand all of these algebraic structures to be graded, and we say that a coalgebra is connected if $C_i=0$ for $i<0$ and $\epz: C_0\rtarr \bZ_2$ is an isomorphism. When considering the Hurewicz homomorphism of $\pi_*(TO)$, we shall need the following observation. \begin{lem} Let $C$ be a connected coalgebra and $V$ be a vector space. An element $y\in C\ten V$ satisfies $(\ps\ten\id)(y) = 1\ten y$ if and only if $y\in C_0\ten V\iso V$. \end{lem} If $V$ is a $C$-comodule with coaction $\nu: V\rtarr C\ten V$, then $\nu$ is a morphism of $C$-comodules. Therefore the coaction maps $\ga$ above are maps of $A_*$-comodules for any space $X$ or prespectrum $T$. We also need the following observation, which is implied by the Cartan formula. \begin{lem} If $T$ is an associative ring prespectrum, then $\ga: H_*(T)\rtarr A_*\ten H_*(T)$ is a homomorphism of algebras. \end{lem} The lemma above on Steenrod operations in $H^*(\bR P^{\infty})$ dualizes as follows. \begin{lem} Write the coaction $\ga: H_*(\bR P^{\infty})\rtarr A_*\ten H_*(\bR P^{\infty})$ in the form $\ga(x_i) = \sum_j a_{i,j}\ten x_j$. Then $$ a_{i,1}= \left\{ \begin{array}{ll} \xi_r & \mbox{if $i=2^r$ for some $r\geq 1$}\\ 0 & \mbox{otherwise.} \end{array} \right. $$ \end{lem} Note that $a_{i,i}=1$, dualizing $Sq^0(\al^i)=\al^i$. Armed with this information, we return to the study of the algebra $H_*(TO)$. We know that it is isomorphic to $H_*(BO)$, but the crux of the matter is to redescribe it in terms of $A_*$. \begin{thm} Let $N_*$ be the algebra defined abstractly by $$N_*=\bZ_2[u_i|i>1 \tand i\neq 2^r-1],$$ where $\deg u_i = i$. Define a homomorphism of algebras $f: H_*(TO)\rtarr N_*$ by $$ f(a_i)= \left\{ \begin{array}{ll} u_i & \mbox{if $i$ is not of the form $2^r-1$}\\ 0 & \mbox{if $i=2^r-1$.} \end{array} \right. $$ Then the composite $$g: H_*(TO)\overto{\ga} A_*\ten H_*(TO) \overto{\id\ten f} A_*\ten N_*$$ is an isomorphism of both $A$-comodules and $\bZ_2$-algebras. \end{thm} \begin{proof} It is clear from things already stated that $g$ is a map of both $A$-comodules and $\bZ_2$-algebras. We must prove that it is an isomorphism. Its source and target are both polynomial algebras with one generator of degree $i$ for each $i\geq 1$, hence it suffices to show that $g$ takes generators to generators. Recall that $a_i=j_*(x_{i+1})$. This allows us to compute $\ga(a_i)$. Modulo terms that are decomposable in the algebra $A_*\ten H_*(TO)$, we find $$ \ga(a_i)\equiv \left\{ \begin{array}{ll} 1\ten a_i & \mbox{if $i$ is not of the form $2^r-1$}\\ \xi_r\ten 1 + 1\ten a_{2^r-1} & \mbox{if $i=2^r-1$.} \end{array} \right. $$ Applying $\id\ten f$ to these elements, we obtain $1\ten u_i$ in the first case and $\xi_r\ten 1$ in the second case. \end{proof} Now consider the Hurewicz homomorphism $h: \pi_*(T)\rtarr H_*(T)$ of a prespectrum $T$. We have the following observation, which is a direct consequence of the definition of the Hurewicz homomorphism and the fact that $Sq^i = 0$ for $i>0$ in the cohomology of spheres. \begin{lem} For $x\in\pi_*(T)$, $\ga(h(x))=1\ten h(x)$. \end{lem} Therefore, identifying $N_*$ as the subalgebra $\bZ_2\ten N_*$ of $A_*\ten N_*$, we see that $g\com h$ maps $\pi_*(TO)$ to $N_*$. We shall prove the following result in \S6 and so complete the proof of Thom's theorem. \begin{thm} $h: \pi_*(TO)\rtarr H_*(TO)$ is a monomorphism and $g\com h$ maps $\pi_*(TO)$ isomorphically onto $N_*$. \end{thm} \section{The relationship to Stiefel-Whitney numbers} We shall prove that a smooth closed $n$-manifold $M$ is a boundary if and only if all of its normal Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!normal} are zero. Polynomials in the Stiefel-Whitney classes are elements of $H^*(BO)$. We have seen that the normal Stiefel-Whitney numbers of a boundary are zero, and it follows that cobordant manifolds have the same normal Stiefel-Whitney numbers. The assignment of Stiefel-Whitney numbers to corbordism classes of $n$-manifolds specifies a homomorphism $$\#: H^n(BO)\ten \sN_n \rtarr \bZ_2.$$ We claim that the following diagram is commutative: $$\diagram H^n(BO)\ten \sN_n \rto^(0.45){\id\ten\al} \dto_{\#} & H^n(BO)\ten \pi_n(TO) \rto^{\id\ten h} & H^n(BO)\ten H_n(TO) \dto^{\id\ten \PH} \\ \bZ_2 & & H^n(BO)\ten H_n(BO). \llto_{\langle \, , \, \rangle}\\ \enddiagram$$ To say that all normal Stiefel-Whitney numbers of $M$ are zero is to say that $w\#[M]=0$ for all $w\in H^n(BO)$. Granted the commutativity of the diagram, this is the same as to say that $\langle w,(\PH\com h\com \al)([M])\rangle = 0$ for all $w\in H^n(BO)$. Since $\langle \, , \, \rangle$ is the evaluation pairing of dual vector spaces, this implies that $(\PH\com h\com \al)([M])=0$. Since $\PH$ and $\al$ are isomorphisms and $h$ is a monomorphism, this implies that $[M]=0$ and thus that $M$ is a boundary. Thus we need only prove that the diagram is commutative. Embed $M$ in $\bR^{n+q}$ with normal bundle $\nu$ and let $f: M\rtarr BO(q)$ classify $\nu$. Then $\al([M])$ is represented by the composite $S^{n+q}\overto{t} T\nu\overto{Tf} TO(q)$. In homology, we have the commutative diagram $$\diagram \tilde{H}_{n+q}(S^{n+q}) \rto^{t_*} & \tilde{H}_{n+q}(T\nu) \rto^{(Tf)_*} \dto^{\PH} & \tilde{H}_{n+q}(TO(q)) \dto^{\PH} \\ & H_n(M) \rto_{f_*} & H_n(BO(q)).\\ \enddiagram$$ Let $i_{n+q}\in \tilde{H}_{n+q}(S^{n+q})$ be the fundamental class. By the diagram and the definitions of $\al$ and the Hurewicz homomorphism, $$ (f_*\com \PH\com t_*)(i_{n+q}) = (\PH\com (Tf)_*\com t_*)(i_{n+q}) = (\PH\com h\com \al)([M]) \in H_n(BO(q)).$$ Let $z=(\PH\com t_*)(i_{n+q})\in H_n(M)$. We claim that $z$ is the fundamental class. Granting the claim, it follows immediately that, for $w\in H^n(BO(q))$, \begin{eqnarray*} w\# [M] = \langle w(\nu), z\rangle & = & \langle (f^*w(\ga_q)),(\PH\com t_*)(i_{n+q}) \rangle \\ & = & \langle w(\ga_q), (f_*\com \PH\com t_*)(i_{n+q})\rangle \\ & = & \langle w(\ga_q), (\PH\com h\com \al)([M])\rangle. \end{eqnarray*} Thus we are reduced to proving the claim. It suffices to show that $z$ maps to a generator of $H_n(M,M-x)$ for each $x\in M$. Since we must deal with pairs, it is convenient to use the homeomorphism between $T\nu$ and the quotient $D(\nu)/S(\nu)$ of the unit disk bundle by the unit sphere bundle. Recall that we have a relative cap product $$\cap: H^q(D(\nu),S(\nu))\ten H_{i+q}(D(\nu),S(\nu))\rtarr H_i(D(\nu)).$$ Letting $p:D(\nu)\rtarr M$ be the projection, which of course is a homotopy equivalence, we find that the homology Thom isomorphism $$\PH: H_{i+q}(D(\nu),S(\nu))\rtarr H_i(M)$$ is given by the explicit formula $$ \PH(a) = p_*(\mu \cap a).$$ Let $x\in U\subset M$, where $U\iso \bR^n$. Let $D(U)$ and $S(U)$ be the inverse images in $U$ of the unit disk and unit sphere in $\bR^n$ and let $V=D(U)-S(U)$. Since $D(U)$ is contractible, $\nu|_{D(U)}$ is trivial and thus isomorphic to $D(U)\times D^q$. Write $$\pa(D(U)\times D^q) = (D(U)\times S^{q-1})\cup (S(U)\times D^q)$$ and observe that we obtain a homotopy equivalence $$t: S^{n+q} \rtarr (D(U)\times D^q)/\pa (D(U)\times D^q)\iso S^{n+q}$$ by letting $t$ be the quotient map on the restriction of the tubular neighborhood of $\nu$ to $D(\nu|_{D(U)})$ and letting $t$ send the complement of this restriction to the basepoint. Interpreting $t: S^{n+q}\rtarr D(\nu)/S(\nu)$ similarly, we obtain the following commutative diagram: \begin{small} $$\diagram \tilde{H}_{n+q}(S^{n+q}) \rto^(0.3){t_*}_(0.3){\iso} \ddto_{t_*} & H_{n+q}(D(U)\times D^q,\pa(D(U)\times D^q)) \rto^(0.63){\PH}_(0.63){\iso} \dto & H_n(D(U),S(U)) \dto^{\iso} \\ & H_{n+q}(D(\nu),S(\nu)\cup D(\nu|_{M-V})) \rto^(0.6){\PH} & H_n(M,M-V) \dto^{\iso} \\ H_{n+q}(D(\nu),S(\nu))\urto \rto_(0.55){\PH} & H_n(M) \urto \rto & H_n(M,M-x).\\ \enddiagram$$ \end{small} The unlabeled arrows are induced by inclusions, and the right vertical arrows are excision isomorphisms. The maps $\PH$ are of the general form $\PH(a)=p_*(\mu\cap a)$. For the top map $\PH$, $\mu\in H_{n+q}(D(\nu|_{D(U)}),S(\nu|_{D(U)}))\iso H_{n+q}(S^{n+q})$, and, up to evident isomorphisms, $\PH$ is just the inverse of the suspension isomorphism $\tilde{H}_n(S^n) \rtarr \tilde{H}_{n+q}(S^{n+q})$. The diagram shows that $z$ maps to a generator of $H_n(M,M-x)$, as claimed. \section{Spectra and the computation of $\pi_*(TO) =\pi_*(MO)$} We must still prove that $h:\pi_*(TO)\rtarr H_*(TO)$ is a monomorphism and that $g\com h$ maps $\pi_*(TO)$ isomorphically onto $N_*$. Write $N$ for the dual vector space of $N_*$. (Of course, $N$ is a coalgebra, but that is not important for this part of our work.) Remember that the Steenrod algebra $A$ is dual to $A_*$ and that $A\iso H^*(H\bZ_2)$. The dual of $g:H_*(TO)\rtarr A_*\ten N_*$ is an isomorphism of $A$-modules (and of coalgebras) $g^*: A\ten N\rtarr H^*(TO)$. Thus, if we choose a basis $\sset{y_i}$ for $N$, where $\deg\,y_i = n_i$ say, then $H^*(TO)$ is the free graded $A$-module on the basis $\sset{y_i}$. At this point, we engage in a conceptual thought exercise. We think of prespectra as ``stable objects''\index{stable objects} that have associated homotopy, homology, and cohomology groups. Imagine that we have a good category of stable objects, analogous to the category of based spaces, that is equipped with all of the constructions that we have on based spaces: wedges (= coproducts), colimits, products, limits, suspensions, loops, homotopies, cofiber sequences, fiber sequences, smash products, function objects, and so forth. Let us call the stable objects in our imagined category ``spectra''\index{spectrum} and call the category of such objects $\sS$.\index{S@$\sS$} We have in mind an analogy with the notions of presheaf and sheaf. Whatever spectra are, there must be a way of constructing a spectrum from a prespectrum without changing its homotopy, homology, and cohomology groups. In turn, a based space $X$ determines the prespectrum $\SI^{\infty} X=\sset{\SI^nX}$. The homology and cohomology groups of $\SI^{\infty} X$ are the (reduced) homology and cohomology groups of $X$; the homotopy groups of $\SI^{\infty} X$ are the stable homotopy groups of $X$. Because homotopy groups, homology groups, and cohomology groups on based spaces satisfy the weak equivalence axiom, the real domain of definition of these invariants is the category $\bar{h}\sT$ that is obtained from the homotopy category $h\sT$ of based spaces by adjoining inverses to the weak equivalences. This category is equivalent to the homotopy category $h\sC$ of based CW complexes. Explicitly, the morphisms from $X$ to $Y$ in $\bar{h}\sT$ can be defined to be the based homotopy classes of maps $\GA X\rtarr \GA Y$, where $\GA X$ and $\GA Y$ are CW approximations of $X$ and $Y$. Composition is defined in the evident way. Continuing our thought exercise, we can form the homotopy category $h\sS$ of spectra and can define homotopy groups in terms of homotopy classes of maps from sphere spectra to spectra. Reflection on the periodic nature of $K$-theory suggests that we should define sphere spectra of negative dimension and define homotopy groups $\pi_q(X)$ for all integers $q$. We say that a map of spectra is a weak equivalence if it induces an isomorphism on homotopy groups. We can form the ``stable category''\index{stable category} $\bar{h}\sS$ from $h\sS$ exactly as we formed the category $\bar{h}\sT$ from $h\sT$. That is, we develop a theory of CW spectra using sphere spectra as the domains of attaching maps. The Whitehead and cellular approximation theorems hold, and every spectrum $X$ admits a CW approximation $\GA X\rtarr X$. We define the set $[X,Y]$ of morphisms $X\rtarr Y$ in $\bar{h}\sS$ to be the set of homotopy classes of maps $\GA X\rtarr \GA Y$. This is a {\em stable} category in the sense that the functor $\SI: \bar{h}\sS \rtarr \bar{h}\sS$ is an equivalence of categories. More explicitly, the natural maps $X\rtarr \OM\SI X$ and $\SI\OM X\rtarr X$ are isomorphisms in $\bar{h}\sS$. In particular, up to isomorphism, every object in the category $\bar{h}\sS$ is a suspension, hence a double suspension. This implies that each $[X,Y]$ is an Abelian group and composition is bilinear. Moreover, for any map $f: X\rtarr Y$, the canonical map $Ff\rtarr \OM Cf$ and its adjoint $\SI Ff\rtarr Cf$ (see Chapter 8 \S7) are also isomorphisms in $\bar{h}\sS$, so that cofiber sequences and fiber sequences are equivalent. Therefore cofiber sequences give rise to long exact sequences of homotopy groups. The homotopy groups of wedges and products of spectra are given by $$\pi_*(\textstyle{\bigvee}_i\, X_i) = \textstyle{\sum}_i\, \pi_*(X_i) \tand \pi_*(\textstyle{\prod}_i\, X_i)=\textstyle{\prod}_i\, \pi_*(X_i).$$ Therefore, if only finitely many $\pi_q(X_i)$ are non-zero for each $q$, then the natural map $\bigvee_i\, X_i\rtarr \prod_i\, X_i$ is an isomorphism. We have homology groups and cohomology groups defined on $\bar{h}\sS$. A spectrum $E$ represents a homology theory\index{homology theory!on spectra} $E_*$ and a cohomology theory\index{cohomology theory!on spectra} $E^*$ specified in terms of smash products and function spectra by $$E_q(X) =\pi_q(X\sma E) \ \tand \ E^q(X) = \pi_{-q}F(X,E) \iso [X,\SI^qE].$$ Verifications of the exactness, suspension, additivity, and weak equivalence axioms are immediate from the properties of the category $\bar{h}\sS$. Moreover, every homology or cohomology theory on $\bar{h}\sS$ is so represented by some spectrum $E$. As will become clear later, $\OM$-prespectra are more like spectra than general prespectra, and we continue to write $H\pi$ for the ``Eilenberg-Mac\,Lane spectrum'' \index{Eilenberg-Mac\,Lane spectrum} that represents ordinary cohomology with coefficients in $\pi$. Its only non-zero homotopy group is $\pi_0(H\pi)=\pi$, and the Hurewicz homomorphism maps this group isomorphically onto $H_0(H\pi;\bZ)$. When $\pi=\bZ_2$, the natural map $H_0(H\bZ_2;\bZ)\rtarr H_0(H\bZ_2;\bZ_2)$ is also an isomorphism. Returning to our motivating example, we write $MO$\index{MO@$MO$} for the ``Thom spectrum''\index{Thom spectrum} that arises from the Thom prespectrum $TO$. The reader may sympathize with a student who claimed that $MO$ stands for ``Mythical Object.''\index{Mythical Object} We may choose a map $\bar{y}_i: MO \rtarr \SI^{n_i}H\bZ_2$ that represents the element $y_i$. Define $K(N_*)$ to be the wedge of a copy of $\SI^{n_i}H\bZ_2$ for each basis element $y_i$ and note that $K(N_*)$ is isomorphic in $\bar{h}\sS$ to the product of a copy of $\SI^{n_i}H\bZ_2$ for each $y_i$. We think of $K(N_*)$ as a ``generalized Eilenberg-Mac\,Lane spectrum.'' It satisfies $\pi_*(K(N_*))\iso N_*$ (as Abelian groups and so as $\bZ_2$-vector spaces), and the mod $2$ Hurewicz homomorphism $h: \pi_*(K(N_*))\rtarr H_*(K(N_*))$ is a monomorphism. Using the $\bar{y}_i$ as coordinates, we obtain a map $$\om: MO\rtarr \textstyle{\prod}_i\, \SI^{n_i} H\bZ_2 \htp K(N_*).$$ The induced map $\om^*$ on mod $2$ cohomology is an isomorphism of $A$-modules: $H^*(MO)$ and $H^*(K(N_*))$ are free $A$-modules, and we have defined $\om$ so that $\om^*$ sends basis elements to basis elements. Therefore the induced map on homology groups is an isomorphism. Here we are using mod $2$ homology, but it can be deduced from the fact that both $\pi_*(MO)$ and $\pi_*(K(N_*))$ are $\bZ_2$-vector spaces that $\om$ induces an isomorphism on integral homology groups. Therefore the integral homology groups of $C\om$ are zero. By the Hurewicz theorem in $\bar{h}\sS$, the homotopy groups of $C\om$ are also zero. Therefore $\om$ induces an isomorphism of homotopy groups. That is, $\om$ is an isomorphism in $\bar{h}\sS$. Therefore $\pi_*(MO)\iso N_*$ and the Hurewicz homomorphism $h:\pi_*(MO)\rtarr H_*(MO)$ is a monomorphism. It follows that $g\com h:\pi_*(MO)\rtarr N_*$ is an isomorphism since it is a monomorphism between vector spaces of the same finite dimension in each degree. \section{An introduction to the stable category} To give content to the argument just sketched, we should construct a good category of spectra. In fact, no such category was available when Thom first proved his theorem in 1960. With motivation from the introduction of $K$-theory and cobordism, a good stable category was constructed by Boardman (unpublished) around 1964 and an exposition of his category was given by Adams soon after. However, these early constructions were far more primitive than our outline suggests. While they gave a satisfactory stable category, the underlying category of spectra did not have products, limits, and function objects, and its smash product was not associative, commutative, or unital. In fact, a fully satisfactory category of spectra was not constructed until 1995. We give a few definitions to indicate what is involved. \begin{defn} A spectrum\index{spectrum} $E$ is a prespectrum such that the adjoints $\tilde{\si}: E_n\rtarr \OM E_{n+1}$ of the structure maps $\si: \SI E_n \rtarr E_{n+1}$ are {\em homeomorphisms}. A map $f: T\rtarr T'$ of prespectra is a sequence of maps $f_n: T_n\rtarr T'_n$ such that $\si'_n\com \SI f_n = f_{n+1}\com \si_n$ for all $n$. A map $f:E\rtarr E'$ of spectra is a map between $E$ and $E'$ regarded as prespectra. \end{defn} We have a forgetful functor from the category $\sS$\index{S@$\sS$} of spectra to the category $\sP$\index{P@$\sP$} of prespectra. It has a left adjoint $L:\sP\rtarr \sS$. In $\sP$, we define wedges, colimits, products, and limits spacewise. For example, $(T\wed T')_n = T_n\wed T'_n$, with the evident structure maps. We define wedges and colimits of spectra by first performing the construction on the prespectrum level and then applying the functor $L$. If we start with spectra and construct products or limits spacewise, then the result is again a spectrum; that is, limits of spectra are the limits of their underlying prespectra. Thus the category $\sS$ is complete and cocomplete. Similarly, we define the smash product $T\sma X$ and function prespectrum $F(X,T)$ of a based space $X$ and a prespectrum $T$ spacewise. For a spectrum $E$, we define $E\sma X$ by applying $L$ to the prespectrum level construction; the prespectrum $F(X,E)$ is already a spectrum. We now have cylinders $E\sma I_+$ and thus can define homotopies between maps of spectra. Similarly we have cones $CE=E\sma I$ (where $I$ has basepoint $1$), suspensions $\SI E=E\sma S^1$, path spectra $F(I,E)$ (where $I$ has basepoint $0$), and loop spectra $\OM E =F(S^1,E)$. The development of cofiber and fiber sequences proceeds exactly as for based spaces. The left adjoint $L$ can easily be described explicitly on those prespectra $T$ whose adjoint structure maps $\tilde{\si}_n: T_n\rtarr \OM T_{n+1}$ are inclusions: we define $(LT)_n$ to be the union of the expanding sequence $$T_n \overto{\tilde{\si}_n} \OM T_{n+1} \overto{\OM\tilde{\si}_{n+1}} \OM^2 T_{n+2} \rtarr \cdots.$$ We then have $$\OM (LT)_{n+1} =\OM(\bigcup \OM^qT_{n+1+q}) \iso \bigcup \OM^{q+1}T_{n+q+1} \iso (LT)_n.$$ We have an evident map of prespectra $\la: T\rtarr LT$, and a comparison of colimits shows (by a cofinality argument) that $\la$ induces isomorphisms on homotopy and homology groups. The essential point is that homotopy and homology commute with colimits. It is not true that cohomology converts colimits to limits in general, because of ${\lim}^1$ error terms, and this is one reason that our definition of the cohomology of prespectra via limits is inappropriate except under restrictions that guarantee the vanishing of ${\lim}^1$ terms. Observe that there is no problem in the case of $\OM$-prespectra, for which $\la$ is a spacewise weak equivalence. For a based space $X$, we define the suspension spectrum\index{suspension spectrum} $\SI^{\infty}X$ by applying $L$ to the suspension prespectrum $\SI^{\infty} X =\sset{\SI^nX}$. The inclusion condition is satisfied in this case. We define $QX=\cup \OM^q\SI^q X$,\index{QX@$QX$} and we find that the $n$th space of $\SI^{\infty} X$ is $Q\SI^n X$. It should be apparent that the homotopy groups of the space $QX$ are the stable homotopy groups of $X$. The adjoint structure maps of the Thom prespectrum $TO$ are also inclusions, and our mythical object is $MO=L TO$.\index{MO@$MO$} In general, for a prespectrum $T$, we can apply an iterated mapping cylinder construction to define a spacewise equivalent prespectrum $KT$ whose adjoint structure maps are inclusions. The prespectrum level homotopy, homology, and cohomology groups of $KT$ are isomorphic to those of $T$. Thus, if we have a prespectrum $T$ whose invariants we are interested in, such as an Eilenberg-Mac\,Lane $\OM$-prespectrum or the $K$-theory $\OM$-prespectrum, then we can construct a spectrum $LKT$ that has the same invariants. For a based space $X$ and $q\geq 0$, we construct a prespectrum $\SI^{\infty}_qX$ whose $n$th space is a point for $n 0$. The definition is appropriate since $\SI S^q \iso S^{q+1}$ for all integers $q$. We can now define homotopy groups in the obvious way. For example, the homotopy groups of the $K$-theory spectrum are $\bZ$ for every even integer and zero for every odd integer. From here, we can go on to define CW spectra in very much the same way that we defined CW complexes, and we can fill in the rest of the outline in the previous section. The real work involves the smash product of spectra, but this does not belong in our rapid course. While there is a good deal of foundational work involved, there is also considerable payoff in explicit concrete calculations, as the computation of $\pi_*(MO)$ well illustrates. With the hope that this glimpse into the world of stable homotopy theory has whetted the reader's appetite for more, we will end at this starting point. \clearpage \thispagestyle{empty} \chapter*{Suggestions for further reading} \setcounter{section}{0} Rather than attempt a complete bibliography, I will give a number of basic references. I will begin with historical references and textbooks. I will then give references for specific topics, more or less in the order in which topics appear in the text. Where material has been collected in one or another book, I have often referred to such books rather than to original articles. However, the importance and quality of exposition of some of the original sources often make them still to be preferred today. The subject in its earlier days was blessed with some of the finest expositors of mathematics, for example Steenrod, Serre, Milnor, and Adams. Some of the references are intended to give historical perspective, some are classical papers in the subject, some are follow-ups to material in the text, and some give an idea of the current state of the subject. In fact, many major parts of algebraic topology are nowhere mentioned in any of the existing textbooks, although several were well established by the mid-1970s. I will indicate particularly accessible references for some of them; the reader can find more of the original references in the sources given. \section{A classic book and historical references} The axioms for homology and cohomology theories were set out in the classic: \vspace{1mm} \noindent {\em S. Eilenberg and N. Steenrod. Foundations of algebraic topology.} Princeton University Press. 1952. \vspace{1.3mm} I believe the only historical monograph on the subject is: \vspace{1mm} \noindent {\em J. Dieudonn\'e. A history of algebraic and differential topology, 1900--1960.} Birk\-h\"auser. 1989. \vspace{1.3mm} A large collection of historical essays will appear soon: \vspace{1mm} \noindent {\em I.M. James, editor. The history of topology.} Elsevier Science. To appear. \vspace{1.3mm} Among the contributions, I will advertise one of my own, available on the web: \vspace{1mm} \noindent {\em J.P. May. Stable algebraic topology, 1945--1966.} http://hopf.math.purdue.edu \section{Textbooks in algebraic topology and homotopy theory} These are ordered roughly chronologically (although this is obscured by the fact that the most recent editions or versions are cited). I have included only those texts that I have looked at myself, that are at least at the level of the more elementary chapters here, and that offer significant individuality of treatment. There are many other textbooks in algebraic topology. \vspace{1mm} Two classic early textbooks: \vspace{1.3mm} \noindent {\em P.J. Hilton and S. Wylie. Homology theory.} Cambridge University Press. 1960. \vspace{1mm} \noindent {\em E. Spanier. Algebraic topology.} McGraw-Hill. 1966. \vspace{1.3mm} An idiosyncratic pre-homology level book giving much material about groupoids: \vspace{1mm} \noindent {\em R. Brown. Topology. A geometric account of general topology, homotopy types, and the fundamental groupoid.} Second edition. Ellis Horwood. 1988. \vspace{1.3mm} A homotopical introduction close to the spirit of this book: \vspace{1mm} \noindent {\em B. Gray. Homotopy theory, an introduction to algebraic topology.} Academic Press. 1975. \vspace{1.3mm} The standard current textbooks in basic algebraic topology: \vspace{1mm} \noindent {\em M.J. Greenberg and J. R. Harper. Algebraic topology, a first course.} Benjamin/\linebreak Cummings. 1981. \vspace{1mm} \noindent {\em W.S. Massey. A basic course in algebraic topology.} Springer-Verlag. 1991. \vspace{1mm} \noindent {\em A. Dold. Lectures on algebraic topology.} Reprint of the 1972 edition. Springer-Verlag. 1995. \vspace{1mm} \noindent {\em J.W. Vick. Homology theory; an introduction to algebraic topology.} Second edition. Springer-Verlag. 1994. \vspace{1mm} \noindent {\em J.R. Munkres. Elements of algebraic topology.} Addison Wesley. 1984. \vspace{1mm} \noindent {\em J.J. Rotman. An introduction to algebraic topology.} Springer-Verlag. 1986. \vspace{1mm} \noindent {\em G.E. Bredon. Topology and geometry.} Springer-Verlag. 1993. \vspace{1.3mm} Sadly, the following are still the only more advanced textbooks in the subject: \vspace{1mm} \noindent {\em R.M. Switzer. Algebraic topology. Homotopy and homology.} Springer-Verlag. 1975. \vspace{1mm} \noindent {\em G.\!W. Whitehead. Elements of homotopy theory.} Springer-Verlag. 1978. \section{Books on CW complexes} Two books giving more detailed studies of CW complexes than are found in textbooks (the second giving a little of the theory of compactly generated spaces): \vspace{1mm} \noindent {\em A.T. Lundell and S. Weingram The topology of CW complexes.} Van Nostrand Reinhold. 1969. \vspace{1mm} \noindent {\em R. Fritsch and R.A. Piccinini. Cellular structures in topology.} Cambridge University Press. 1990. \section{Differential forms and Morse theory} Two introductions to algebraic topology starting from de Rham cohomology: \vspace{1mm} \noindent {\em R. Bott and L.\!W. Tu. Differential forms in algebraic topology.} Springer-Verlag. 1982. \vspace{1mm} \noindent {\em I. Madsen and J. Tornehave. From calculus to cohomology. de Rham cohomology and characteristic classes.} Cambridge University Press. 1997. \vspace{1.3mm} The classic reference on Morse theory, with an exposition of the Bott periodicity theorem: \vspace{1mm} \noindent {\em J. Milnor. Morse theory.} Annals of Math. Studies No. 51. Princeton University Press. 1963. \vspace{1.3mm} A modern use of Morse theory for the analytic construction of homology: \vspace{1mm} \noindent {\em M. Schwarz. Morse homology.} Progress in Math. Vol. 111. Birkh\"auser. 1993. %R. Bott. An application of the Morse theory to the topology of Lie-groups. %Bull. Soc. Math. France 84(1956), 251-281. %R. Bott. The stable homotopy of the classical groups. Annals of Math. 70(1959), 313-337. %R. Bott. Quelques remarques sur les th\'eor\`emes de periodicit\'e de topology. Bull. Soc. %Math. France 87(1959), 293-310. %M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles. %Acta Math. 112(1964), 229-247. \section{Equivariant algebraic topology} Two good basic references on equivariant algebraic topology, classically called the theory of transformation groups (see also \S\S16, 21 below): \vspace{1mm} \noindent {\em G. Bredon. Introduction to compact transformation groups.} Academic Press. 1972. \vspace{1mm} \noindent {\em T. tom Dieck. Transformation groups.} Walter de Gruyter. 1987. \vspace{1.3mm} A more advanced book, a precursor to much recent work in the area: \vspace{1mm} \noindent {\em T. tom Dieck. Transformation groups and representation theory.} Lecture Notes in Mathematics Vol. 766. Springer-Verlag. 1979. \section{Category theory and homological algebra} A revision of the following classic on basic category theory is in preparation: \vspace{1mm} \noindent {\em S. Mac\,Lane. Categories for the working mathematician.} Springer-Verlag. 1971. \vspace{1.3mm} Two classical treatments and a good modern treatment of homological algebra: \vspace{1mm} \noindent {\em H. Cartan and S. Eilenberg. Homological algebra.} Princeton University Press. 1956. \vspace{1mm} \noindent {\em S. MacLane. Homology.} Springer-Verlag. 1963. \vspace{1mm} \noindent {\em C.A. Weibel. An introduction to homological algebra.} Cambridge University Press. 1994. \section{Simplicial sets in algebraic topology} Two older treatments and a comprehensive modern treatment: \vspace{1mm} \noindent {\em P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory.} Springer-Verlag. 1967. \vspace{1mm} \noindent {\em J.P. May. Simplicial objects in algebraic topology.} D. Van Nostrand 1967; reprinted by the University of Chicago Press 1982 and 1992. \vspace{1mm} \noindent {\em P.G. Goerss and J.F. Jardine. Simplicial homotopy theory.} Birkh\"auser. To appear. \section{The Serre spectral sequence and Serre class theory} Two classic papers of Serre: \vspace{1mm} \noindent {\em J.-P. Serre. Homologie singuli\'ere des espaces fibr\'es. Applications.} Annals of Math. (2)54(1951), 425--505. \vspace{1mm} \noindent {\em J.-P. Serre. Groupes d'homotopie et classes de groupes ab\'eliens.} Annals of Math. (2)58(1953), 198--232. \vspace{1.3mm} A nice exposition of some basic homotopy theory and of Serre's work: \vspace{1mm} \noindent {\em S.-T. Hu. Homotopy theory.} Academic Press. 1959. \vspace{1.3mm} Many of the textbooks cited in \S2 also treat the Serre spectral sequence. \section{The Eilenberg-Moore spectral sequence} There are other important spectral sequences in the context of fibrations, mainly due to Eilenberg and Moore. Three references: \vspace{1mm} \noindent {\em S. Eilenberg and J.C. Moore. Homology and fibrations, I.} Comm. Math. Helv. 40(1966), 199--236. \vspace{1mm} \noindent {\em L. Smith. Homological algebra and the Eilenberg-Moore spectral sequences.} Trans. Amer. Math. Soc. 129(1967), 58--93. \vspace{1mm} \noindent {\em V.K.A.M. Gugenheim and J.P. May. On the theory and applications of differential torsion products.} Memoirs Amer. Math. Soc. No. 142. 1974. \vspace{1.3mm} There is a useful guidebook to spectral sequences: \vspace{1mm} \noindent {\em J. McCleary. User's guide to spectral sequences.} Publish or Perish. 1985. \section{Cohomology operations} A compendium of the work of Steenrod and others on the construction and analysis of the Steenrod operations: \vspace{1mm} \noindent {\em N.E. Steenrod and D.B.A. Epstein. Cohomology operations.} Annals of Math. Studies No. 50. Princeton University Press. 1962. \vspace{1.3mm} A classic paper that first formalized cohomology operations, among other things: \vspace{1mm} \noindent {\em J.-P. Serre. Cohomologie modulo $2$ des complexes d'Eilenberg-Mac\,Lane.} Comm. Math. Helv. 27(1953), 198--232. \vspace{1.3mm} A general treatment of Steenrod-like operations: \vspace{1mm} \noindent {\em J.P. May. A general algebraic approach to Steenrod operations.} In Lecture Notes in Mathematics Vol. 168, 153--231. Springer-Verlag. 1970. \vspace{1.3mm} A nice book on mod $2$ Steenrod operations and the Adams spectral sequence: \vspace{1mm} \noindent {\em R. Mosher and M. Tangora. Cohomology operations and applications in homotopy theory.} Harper and Row. 1968. \section{Vector bundles} A classic and a more recent standard treatment that includes $K$-theory: \vspace{1mm} \noindent {\em N.E. Steenrod. Topology of fibre bundles.} Princeton University Press. 1951. Fifth printing, 1965. \vspace{1mm} \noindent {\em D. Husemoller. Fibre bundles.} Springer-Verlag. 1966. Third edition, 1994. \vspace{1.3mm} A general treatment of classification theorems for bundles and fibrations: \vspace{1mm} \noindent {\em J.P. May. Classifying spaces and fibrations.} Memoirs Amer. Math. Soc. No. 155. 1975. \section{Characteristic classes} The classic introduction to characteristic classes: \vspace{1mm} \noindent {\em J. Milnor and J.D. Stasheff. Characteristic classes.} Annals of Math. Studies No. 76. Princeton University Press. 1974. \vspace{1.3mm} A good reference for the basic calculations of characteristic classes: \vspace{1mm} \noindent {\em A. Borel. Topology of Lie groups and characteristic classes.} Bull. Amer. Math. Soc. 61(1955), 297--432. \vspace{1.3mm} Two proofs of the Bott periodicity theorem that only use standard techniques of algebraic topology, starting from characteristic class calculations: \vspace{1mm} \noindent {\em H. Cartan et al. P\'eriodicit\'e des groupes d'homotopie stables des groupes classiques, d'apr\`es Bott.} S\'eminaire Henri Cartan, 1959/60. Ecole Normale Sup\'erieure. Paris. \vspace{1mm} \noindent {\em E. Dyer and R.K. Lashof. A topological proof of the Bott periodicity theorems.} Ann. Mat. Pure Appl. (4)54(1961), 231--254. \section{$K$-theory} %M.F. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces, in Differential %Geometry. Amer. Math. Soc. Proc. Symp. Pure Math 3(1961), 7--38. Two classical lecture notes on $K$-theory: \vspace{1mm} \noindent {\em R. Bott. Lectures on $K(X)$.} W.A. Benjamin. 1969. \vspace{1mm} This includes a reprint of perhaps the most accessible proof of the complex case of the Bott periodicity theorem, namely: \vspace{1mm} \noindent {\em M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles.} Acta Math. 112(1994), 229--247. \vspace{1.3mm} \noindent {\em M.F. Atiyah. $K$-theory.} Notes by D.W. Anderson. Second Edition. Addison-Wesley. 1967. \vspace{1mm} This includes reprints of two classic papers of Atiyah, one that relates Adams operations in $K$-theory to Steenrod operations in cohomology and another that sheds insight on the relationship between real and complex $K$-theory: \vspace{1mm} \noindent {\em M.F. Atiyah. Power operations in $K$-theory.} Quart. J. Math. (Oxford) (2)17(1966), 165--193. \vspace{1mm} \noindent {\em M.F. Atiyah. $K$-theory and reality.} Quart. J. Math. (Oxford) (2)17(1966), 367--386. \vspace{1.3mm} Another classic paper that greatly illuminates real $K$-theory: \vspace{1mm} \noindent {\em M.F. Atiyah, R. Bott, and A. Shapiro. Clifford algebras.} Topology 3(1964), suppl. 1, 3--38. \vspace{1.3mm} A more recent book on $K$-theory: \noindent {\em M. Karoubi. $K$-theory.} Springer-Verlag. 1978. \vspace{1.3mm} Some basic papers of Adams and Adams and Atiyah giving applications of $K$-theory: \vspace{1mm} \noindent {\em J.F. Adams. Vector fields on spheres.} Annals of Math. 75(1962), 603--632. \vspace{1mm} \noindent {\em J.F. Adams. On the groups $J(X)$ I, II, III, and IV.} Topology 2(1963), 181--195; 3(1965), 137-171 and 193--222; 5(1966), 21--71. \vspace{1mm} \noindent {\em J.F. Adams and M.F. Atiyah. $K$-theory and the Hopf invariant.} Quart. J. Math. (Oxford) (2)17(1966), 31--38. \section{Hopf algebras; the Steenrod algebra, Adams spectral sequence} The basic source for the structure theory of (connected) Hopf algebras: \vspace{1mm} \noindent {\em J. Milnor and J.C. Moore. On the structure of Hopf algebras.} Annals of Math. 81(1965), 211--264. \vspace{1.3mm} The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: \vspace{1mm} \noindent {\em J. Milnor. The Steenrod algebra and its dual.} Annals of Math. 67(1958), 150--171. \vspace{1.3mm} Two classic papers of Adams; the first constructs the Adams spectral sequence relating the Steenrod algebra to stable homotopy groups and the second uses secondary cohomology operations to solve the Hopf invariant one problem: \vspace{1mm} \noindent {\em J.F. Adams. On the structure and applications of the Steenrod algebra.} Comm. Math. Helv. 32(1958), 180--214. \vspace{1mm} \noindent {\em J.F. Adams. On the non-existence of elements of Hopf invariant one.} Annals of Math. 72(1960), 20--104. \section{Cobordism} The beautiful classic paper of Thom is still highly recommended: \vspace{1mm} \noindent {\em R. Thom. Quelques propri\'et\'es globals des vari\'et\'es diff\'erentiables.} Comm. Math. Helv. 28(1954), 17--86. \vspace{1.3mm} Thom computed unoriented cobordism. Oriented and complex cobordism came later. In simplest form, the calculations use the Adams spectral sequence: \vspace{1mm} \noindent {\em J. Milnor. On the cobordism ring $\Omega^*$ and a complex analogue.} Amer. J. Math. 82(1960), 505--521. \vspace{1mm} \noindent {\em C.T.C. Wall. A characterization of simple modules over the Steenrod algebra mod $2$.} Topology 1(1962), 249--254. \vspace{1mm} \noindent {\em A. Liulevicius. A proof of Thom's theorem.} Comm. Math. Helv. 37(1962), 121--131. \vspace{1mm} \noindent {\em A. Liulevicius. Notes on homotopy of Thom spectra.} Amer. J. Math. 86(1964), 1--16. \vspace{1.3mm} A very useful compendium of calculations of cobordism groups: \vspace{1mm} \noindent {\em R. Stong. Notes on cobordism theory.} Princeton University Press. 1968. \section{Generalized homology theory and stable homotopy theory} Two classical references, the second of which also gives detailed information about complex cobordism that is of fundamental importance to the subject. \vspace{1mm} \noindent {\em G.W. Whitehead. Generalized homology theories.} Trans. Amer. Math. Soc. 102(1962), 227--283. \vspace{1mm} \noindent {\em J.F. Adams. Stable homotopy and generalised homology.} Chicago Lectures in Mathematics. University of Chicago Press. 1974. Reprinted in 1995. \vspace{1.3mm} An often overlooked but interesting book on the subject: \vspace{1mm} \noindent {\em H.R. Margolis. Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category.} North-Holland. 1983. \vspace{1.3mm} Foundations for equivariant stable homotopy theory are established in: \vspace{1mm} \noindent {\em L.G. Lewis, Jr., J.P. May, and M.Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory.} Lecture Notes in Mathematics Vol. 1213. Springer-Verlag. 1986. \section{Quillen model categories} In the introduction, I alluded to axiomatic treatments of ``homotopy theory.'' Here are the original and two more recent references: \vspace{1mm} \noindent {\em D.G. Quillen. Homotopical algebra.} Lecture Notes in Mathematics Vol. 43. Springer-Verlag. 1967. \vspace{1mm} \noindent {\em W.G. Dwyer and J. Spalinski. Homotopy theories and model categories}. In A handbook of algebraic topology, edited by I.M. James. North-Holland. 1995. \vspace{1.3mm} The cited ``{\em Handbook}'' (over 1300 pages) contains an uneven but very interesting collection of expository articles on a wide variety of topics in algebraic topology. \vspace{1.3mm} \noindent {\em M. Hovey. Model categories.} Amer. Math. Soc. Surveys and Monographs No. 63. 1998. \section{Localization and completion; rational homotopy theory} Since the early 1970s, it has been standard practice in algebraic topology to localize and complete topological spaces, and not just their algebraic invariants, at sets of primes and then to study the subject one prime at a time, or rationally. Two of the basic original references are: \vspace{1mm} \noindent {\em D. Sullivan. The genetics of homotopy theory and the Adams conjecture.} Annals of Math. 100(1974), 1--79. \vspace{1mm} \noindent {\em A.K. Bousfield and D.M. Kan. Homotopy limits, completions, and localizations.} Lecture Notes in Mathematics Vol. 304. Springer-Verlag. 1972. \vspace{1.3mm} A more accessible introduction to localization and a readable recent paper on completion are: \vspace{1mm} \noindent {\em P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups and spaces.} North-Holland. 1975. \vspace{1mm} \noindent {\em F. Morel. Quelques remarques sur la cohomologie modulo $p$ continue des pro-$p$-espaces et les resultats de J. Lannes concernent les espaces fonctionnel Hom$(BV,X)$.} Ann. Sci. Ecole Norm. Sup. (4)26(1993), 309--360. \vspace{1.3mm} When spaces are rationalized, there is a completely algebraic description of the result. The main original reference and a more accessible source are: \vspace{1mm} \noindent {\em D. Sullivan. Infinitesimal computations in topology.} Publ. Math. IHES 47(1978), 269--332. \vspace{1mm} \noindent {\em A.K. Bousfield and V.K.A.M. Gugenheim. On PL de Rham theory and rational homotopy type.} Memoirs Amer. Math. Soc. No. 179. 1976. \section{Infinite loop space theory} Another area well established by the mid-1970s. The following book is a delightful read, with capsule introductions of many topics other than infinite loop space theory, a very pleasant starting place for learning modern algebraic topology: \vspace{1mm} \noindent {\em J.F. Adams. Infinite loop spaces.} Annals of Math. Studies No. 90. Princeton University Press. 1978. \vspace{1.3mm} The following survey article is less easy going, but gives an indication of the applications to high dimensional geometric topology and to algebraic $K$-theory: \vspace{1mm} \noindent {\em J.P. May. Infinite loop space theory.} Bull. Amer. Math. Soc. 83(1977), 456--494. \vspace{1.3mm} Five monographs, each containing a good deal of expository material, that give a variety of theoretical and calculational developments and applications in this area: \vspace{1mm} \noindent {\em J.P. May. The geometry of iterated loop spaces.} Lecture Notes in Mathematics Vol. 271. Springer-Verlag. 1972. \vspace{1mm} \noindent {\em J.M. Boardman and R.M. Vogt. Homotopy invariant algebraic structures on topological spaces.} Lecture Notes in Mathematics Vol. 347. Springer-Verlag. 1973. \vspace{1mm} \noindent {\em F.R. Cohen, T.J. Lada, and J.P. May. The homology of iterated loop spaces.} Lecture Notes in Mathematics Vol. 533. Springer-Verlag. 1976. \vspace{1mm} \noindent {\em J.P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra.} Lecture Notes in Mathematics Vol. 577. Springer-Verlag. 1977. \vspace{1mm} \noindent {\em R. Bruner, J.P. May, J.E. McClure, and M. Steinberger. $H_{\infty}$ ring spectra and their applications.} Lecture Notes in Mathematics Vol. 1176. Springer-Verlag. 1986. \section{Complex cobordism and stable homotopy theory} Adams' book cited in \S16 gives a spectral sequence for the computation of stable homotopy groups in terms of generalized cohomology theories. Starting from complex cobordism and related theories, its use has been central to two waves of major developments in stable homotopy theory. A good exposition for the first wave: \vspace{1mm} \noindent {\em D.C. Ravenel. Complex cobordism and stable homotopy groups of spheres.} Academic Press. 1986. \vspace{1.3mm} The essential original paper and a very nice survey article on the second wave: \vspace{1mm} \noindent {\em E. Devinatz, M.J. Hopkins, and J.H. Smith. Nilpotence and stable homotopy theory.} Annals of Math. 128(1988), 207--242. \vspace{1mm} \noindent {\em M.J. Hopkins. Global methods in homotopy theory.} In Proceedings of the 1985 LMS Symposium on homotopy theory, edited by J.D.S. Jones and E. Rees. London Mathematical Society. 1987. \vspace{1.3mm} The cited {\em Proceedings} contain good introductory survey articles on several other topics in algebraic topology. A larger scale exposition of the second wave is: \vspace{1mm} \noindent {\em D.C. Ravenel. Nilpotence and periodicity in stable homotopy theory.} Annals of Math. Studies No. 128. Princeton University Press. 1992. \section{Follow-ups to this book} There is a leap from the level of this introductory book to that of the most recent work in the subject. One recent book that helps fill the gap is: \vspace{1mm} {\em P. Selick. Introduction to homotopy theory.} Fields Institute Monographs No. 9. American Mathematical Society. 1997. \vspace{1.3mm} There is a recent expository book for the reader who would like to jump right in and see the current state of algebraic topology; although it focuses on equivariant theory, it contains introductions and discussions of many non-equivariant topics: \vspace{1mm} \noindent {\em J.P. May et al. Equivariant homotopy and cohomology theory.} NSF-CBMS Regional Conference Monograph. 1996. \vspace{1.3mm} For the reader of the last section of this book whose appetite has been whetted for more stable homotopy theory, there is an expository article that motivates and explains the properties that a satisfactory category of spectra should have: \vspace{1mm} \noindent {\em J.P. May. Stable algebraic topology and stable topological algebra.} Bulletin London Math. Soc. 30(1998), 225--234. \vspace{1.3mm} The following monograph gives such a category, with many applications; more readable accounts appear in the {\em Handbook} cited in \S17 and in the book just cited: \vspace{1mm} \noindent {\em A. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May, with an appendix by M. Cole. Rings, modules, and algebras in stable homotopy theory.} Amer. Math. Soc. Surveys and Monographs No. 47. 1997. %\input{ConciseRevised.ind} \end{document}