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\title{A Concise Course in Algebraic Topology}
\author{J. P. May}
\let\chapter\section
\begin{document}
\chapter{The fundamental group and some of its applications}
\begin{enumerate}
\item Let $p$ be a polynomial function on $\bC$ which has no root on $S^1$.
Show that the number of roots of $p(z) = 0$ with $|z| < 1$ is the degree
of the map $\hat{p}: S^1\rtarr S^1$ specified by $\hat{p}(z) = p(z)/|p(z)|$.
\item Show that any map $f: S^1 \rtarr S^1$ such that $\text{deg}(f)\neq 1$ has
a fixed point.
\item Let $G$ be a topological group and take its identity element $e$ as its basepoint.
Define the pointwise product of loops $\al$ and $\be$ by $(\al\be)(t) = \al (t)\be (t)$.
Prove that $\al\be$ is equivalent to the composition of paths $\be\cdot\al$.
Deduce that $\pi_1(G,e)$ is Abelian.
\end{enumerate}
\chapter{Categorical language and the van Kampen theorem}
\begin{enumerate}
\item Compute the fundamental group of the two-holed torus (the compact surface of genus $2$
obtained by sewing together two tori along the boundaries of an open disk removed from each).
\item The Klein bottle\index{Klein bottle}
$K$ is the quotient space of $S^1\times I$ obtained by identifying
$(z,0)$ with $(z^{-1},1)$ for $z\in S^1$. Compute $\pi_1(K)$.
\item$*$ Let $X = \{ (p,q)| p \neq -q\}\subset S^n\times S^n$. Define a map $f: S^n\rtarr X$ by
$f(p) = (p,p)$. Prove that $f$ is a homotopy equivalence.
\item Let $\sC$ be a category that has all coproducts and coequalizers. Prove that $\sC$ is
cocomplete (has all colimits). Deduce formally, by use of opposite categories, that a category
that has all products and equalizers is complete.
\end{enumerate}
\chapter{Covering spaces}
In the following two problems, let $G$ be a connected and locally path connected topological
group\index{topological group} with identity element $e$, let $p: H\rtarr G$ be a covering,
and fix $f\in H$ such
that $p(f)=e$. Prove the following. (Hint: Make repeated use of the fundamental theorem
for covering spaces.)
\begin{enumerate}
\item
\begin{enumerate}
\item[(a)] $H$ has a unique continuous product $H\times H\rtarr H$ with identity element $f$
such that $p$ is a homomorphism.
\item[(b)] $H$ is a topological group under this product, and $H$ is Abelian if $G$ is.
\end{enumerate}
\item
\begin{enumerate}
\item[(a)] The kernel $K$ of $p$ is a discrete normal subgroup of $H$.
\item[(b)] In general, any discrete normal subgroup $K$ of a connected topological group $H$
is contained in the center of $H$.
\item[(c)] For $k\in K$, define $t(k): H\rtarr H$ by $t(k)(h) = kh$. Then $k\rtarr t(k)$
specifies an isomorphism between $K$ and the group Aut$(H)$.
\end{enumerate}
\end{enumerate}
Let $X$ and $Y$ be connected, locally path connected, and Hausdorff. A map $f: X\rtarr Y$
is said to be a local homeomorphism\index{local homeomorphism}
if every point of $X$ has an open neighborhood that maps
homeomorphically onto an open set in $Y$.
\begin{enumerate}
\item[3.] Give an example of a surjective local homeomorphism that is not a covering.
\item[4.]* Let $f: X\rtarr Y$ be a local homeomorphism, where $X$ is compact. Prove that $f$
is a (surjective!) covering with finite fibers.
\end{enumerate}
Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$,
define
$$X^H=\sset{x|hx = x\ \text{for all}\ \ h\in H}\subset X;$$
$X^H$ is the $H$-fixed point subspace\index{fixed point space} of $X$.
Topologize the set of functions $G/H\rtarr X$ as the product of copies
of $X$ indexed on the elements of $G/H$, and give the set of $G$-maps
$G/H\rtarr X$ the subspace topology.
\begin{enumerate}
\item[5.] Show that the space of $G$-maps $G/H\rtarr X$
is naturally homeomorphic to $X^H$. In particular, $\sO(G/H,G/K)\iso (G/K)^H$.
\item[6.] Let $X$ be a $G$-space. Show that passage to fixed point spaces,
$G/H \longmapsto X^H$, is the object function of a {\em contravariant}
functor $X^{(-)}: \sO(G)\rtarr \sU$.
\end{enumerate}
\chapter{Graphs}
\begin{enumerate}
\item Let $F$ be a free group on two generators $a$ and $b$. How many subgroups of $F$ have
index $2$? Specify generators for each of these subgroups.
\item Prove that a non-trivial normal subgroup $N$ with infinite index in a free group $F$
cannot be finitely generated.
\item* Essay: Describe a necessary and sufficient condition for a graph to be embeddable
in the plane.
\end{enumerate}
\chapter{Compactly generated spaces}
\begin{enumerate}
\item
\begin{enumerate}
\item[(a)] Any subspace of a weak Hausdorff space is weak Hausdorff.
\item[(b)] Any closed subspace of a $k$-space is a $k$-space.
\item[(c)] An open subset $U$ of a compactly generated space $X$ is compactly
generated if each point has an open neighborhood in $X$ with closure contained
in $U$.
\end{enumerate}
\item* A Tychonoff (or completely regular) space $X$ is a $T_1$-space (points are
closed) such that for each point $x\in X$ and each closed subset $A$ such that
$x\notin A$, there is a function $f: X\rtarr I$ such that $f(x)=0$ and $f(a)=1$ if
$a\in A$. Prove the following (e.g., Kelley, {\em General Topology}).
\begin{enumerate}
\item[(a)] A space is Tychonoff\index{Tychonoff space} if and only if it can be
embedded in a cube (a product of copies of $I$).
\item[(b)] There are Tychonoff spaces that are not $k$-spaces, but every cube is
a compact Hausdorff space.
\end{enumerate}
\item Brief essay: In view of Problems 1 and 2, what should we mean by a ``subspace'' of a
compactly generated space. (We do {\em not} want to restrict the allowable set of subsets.)
\end{enumerate}
\chapter{Cofibrations}
\begin{enumerate}
\item Show that a cofibration $i: A\rtarr X$ is an inclusion with closed image.
\item Let $i: A\rtarr X$ be a cofibration, where $A$ is a contractible space.
Prove that the quotient map $X\rtarr X/A$ is a homotopy equivalence.
\end{enumerate}
\chapter{Fibrations}
\begin{enumerate}
\item Prove the proposition stated in \S5.
\end{enumerate}
\chapter{Based cofiber and fiber sequences}
\begin{enumerate}
\item Prove the two lemmas stated at the end of \S6.
\end{enumerate}
\chapter{Higher homotopy groups}
\begin{enumerate}
\item Show that, if $n\geq 2$, then $\pi_n(X\wed Y)$ is isomorphic to
$$\pi_n(X)\oplus\pi_n(Y)\oplus \pi_{n+1}(X\times Y, X\wed Y).$$
\item Compute $\pi_n(\bR P^n,\bR P^{n-1})$ for $n\geq 2$. Deduce that the quotient map
$$(\bR P^n,\bR P^{n-1})\to (\bR P^n/\bR P^{n-1},*)$$
does not induce an isomorphism of homotopy groups.
\item Compute the homotopy groups of complex projective space $\bC P^n$ in terms
of the homotopy groups of spheres.
\item Verify that the ``Hopf bundles'' are in fact bundles.
\item Show that $\pi_7(S^4)$ contains an element of infinite order.
\item Compute all of the homotopy groups of $\bR P^{\infty}$ and $\bC P^{\infty}$.
\end{enumerate}
\chapter{CW complexes}
\begin{enumerate}
\item Show that complex projective space $\bC P^n$ is a CW complex with one $2q$-cell for each $q$,
$0\leq q\leq n$.
\item Let $X = \sset{x| x = 0 \ \text{or}\ x = 1/n \ \text{for a positive integer $n$}}\subset \bR$.
Show that $X$ does not have the homotopy type of a CW complex.
\item Assume given maps $f: X\rtarr Y$ and $g: Y\rtarr X$ such that $g\com f$ is homotopic to
the identity. (We say that $Y$ ``dominates'' $X$.) Suppose that $Y$ is a CW complex. Prove
that $X$ has the homotopy type of a CW complex.
\end{enumerate}
Define the Euler characteristic\index{Euler characteristic!of a CW complex} $\ch (X)$ of a
finite CW complex $X$ to be the alternating
sum $\sum(-1)^n\ga_n(X)$,
where $\ga_n(X)$ is the number of $n$-cells of $X$. Let $A$ be a subcomplex of a CW complex $X$,
let $Y$ be a CW complex, let $f: A\rtarr Y$ be a cellular map, and let $Y\cup_f X$ be the
pushout of $f$ and the inclusion $A\rtarr X$.
\begin{enumerate}
\item[4.] Show that $Y\cup_f X$ is a CW complex with $Y$ as a subcomplex and
$X/A$ as a quotient complex. Formulate and prove a formula relating the Euler characteristics
$\ch(A)$, $\ch(X)$, $\ch(Y)$, and $\ch(Y\cup_fX)$ when $X$ and $Y$ are finite.
\item[5.]* Think about proving from what we have done so far that $\ch(X)$ depends only on the
homotopy type of $X$, not on its decomposition as a finite CW complex.
\end{enumerate}
\chapter{The homotopy excision and suspension theorems}
\chapter{A little homological algebra}
For a graded vector space $V=\sset{V_n}$ with $V_n = 0$ for all but finitely many $n$ and
with all $V_n$ finite dimensional, define the
Euler characteristic\index{Euler characteristic!of a graded vector space} $\ch(V)$ to be
$\sum(-1)^n \text{dim}\,V_n$.
\begin{enumerate}
\item Let $V'$, $V$, and $V''$ be such graded vector spaces and suppose there is a long
exact sequence
$$\cdots \rtarr V'_n\rtarr V_n\rtarr V''_n\rtarr V'_{n-1}\rtarr \cdots .$$
Prove that $\ch(V) = \ch(V') + \ch (V'')$.
\item If $\sset{V_n, d_n}$ is a chain complex, show that $\ch(V) = \ch(H_*(V))$.
\item Let $0\rtarr \pi\overto{f}\rh\overto{g}\si\rtarr 0$ be an exact sequence of
Abelian groups and let $C$ be a chain complex of flat (= torsion free) Abelian groups. Write
$H_*(C;\pi)=H_*(C\ten \pi)$. Construct a natural long exact sequence
$$ \cdots \rtarr H_q(C;\pi)\overto{f_*} H_q(C;\rh)\overto{g_*} H_q(C;\si)
\overto{\be} H_{q-1}(C;\pi)\rtarr \cdots.$$
The connecting homomorphism $\be$ is called a Bockstein operation.\index{Bockstein operation}
\end{enumerate}
\chapter{Axiomatic and cellular homology theory}
\begin{enumerate}
\item If $X$ is a finite CW complex, show that $\ch(X)=\ch(H_*(X;k))$ for any field $k$.
\item Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let
$f: A\rtarr Y$ be a cellular map. What is the relationship between $H_*(X,A)$ and
$H_*(Y\cup_fX,Y)$? Is there a similar relationship between $\pi_*(X,A)$ and
$\pi_*(Y\cup_fX,Y)$? If not, give a counterexample.
\item Fill in the details of the computation of the differentials on the cellular
chains in the examples in \S5.
\item Compute $H_*(S^m\times S^n)$ for $m\geq 1$ and $n\geq 1$. Convince yourself that
you can do this by use of CW structures, by direct deduction from the axioms, and by
the K\"{u}nneth theorem (for which see Chapter 17).
\item Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group
of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\bC^n$,
$n \geq 2$. Then $\pi\subset S^1$ acts freely on $S^{2n-1}$ via
$$\ze (z_1,\ldots\!, z_n) = (\ze z_1,\ldots\!,\ze z_n). $$
Let $L^n = S^{2n-1}/\pi$ be the orbit space; it is called a lens space and is an odd primary
analogue of $\bR P^n$. The obvious quotient map $S^{2n-1}\rtarr L^n$ is a universal covering.
\begin{enumerate}
\item[(a)] Compute the integral homology of $L^n$, $n\geq 2$, by mimicking the calculation of
$H_*(\bR P^n)$.
\item[(b)] Compute $H_*(L^n;\bZ_p)$, where $\bZ_p = \bZ/p\bZ$.
\end{enumerate}
\end{enumerate}
\chapter{Derivations of properties from the axioms}
\begin{enumerate}
\item Complete the proof that the Mayer-Vietoris sequence is exact.
\end{enumerate}
\chapter{The Hurewicz and uniqueness theorems}
\begin{enumerate}
\item Let $\pi$ be any group. Construct a connected CW complex $K(\pi,1)$ such that
$\pi_1(K(\pi,1))=\pi$ and $\pi_q(K(\pi,1))=0$ for $q\neq 1$.
\item* In Problem 1, it is rarely the case that $K(\pi,1)$ can be constructed as a compact
manifold. What is a necessary condition on $\pi$ for this to happen?
\item Let $n\geq1$ and let $\pi$ be an Abelian group. Construct a connected CW complex
$M(\pi,n)$ such that $\tilde{H}_n(X;\bZ)=\pi$ and $\tilde{H}_q(X;\bZ)=0$ for $q\neq n$.
(Hint: construct $M(\pi,n)$ as the cofiber of a map between wedges of spheres.)
The spaces $M(\pi,n)$ are called Moore spaces.\index{Moore space}
\item Let $n\geq1$ and let $\pi$ be an Abelian group. Construct a connected CW complex
$K(\pi,n)$ such that $\pi_n(X)=\pi$ and $\pi_q(X)=0$ for $q\neq n$. (Hint:
start with $M(\pi,n)$, using the Hurewicz theorem, and kill its higher homotopy groups.)
The spaces $K(\pi,n)$ are called Eilenberg-Mac\,Lane spaces.\index{Eilenberg-Mac\,Lane space}
\item There are familiar spaces that give $K(\bZ,1)$, $K(\bZ_2,1)$, and $K(\bZ,2)$. Name them.
\item Let $X$ be any connected CW complex whose only non-vanishing homotopy
group is $\pi_n(X)\iso \pi$. Construct a homotopy equivalence $K(\pi,n)\rtarr X$,
where $K(\pi,n)$ is the Eilenberg-Mac\,Lane space that you have constructed.
\item* For groups $\pi$ and $\rh$, compute $[K(\pi,n),K(\rh,n)]$; here $[-,-]$
means based homotopy classes of based maps.
\end{enumerate}
\chapter{Singular homology theory}
\begin{enumerate}
\item Let $X$ be a space that satisfies the hypotheses used to construct a universal
cover $\tilde{X}$. Let $\pi=\pi_1(X)$ and consider the action of the group $\pi$
on the space $\tilde{X}$ given by the isomorphism of $\pi$ with Aut$(\tilde{X})$.
Let $A$ be an Abelian group and let $\bZ[\pi]$ act trivially on $A$,
$a\cdot\si = a$ for $\si\in\pi$ and $a\in A$. Do one or both of the following,
and convince yourself that the other choice also works.
\begin{enumerate}
\item[(a)] [Cellular chains] Assume that $X$ is a CW complex. Show that $\tilde{X}$ is a CW complex
such that the action of $\pi$ on $\tilde{X}$ induces an action of the group ring $\bZ[\pi]$
on the cellular chain complex $C_*(\tilde{X})$ such that each $C_q(\tilde{X})$ is a free
$\bZ[\pi]$-module and $$C_*(X;A)\iso A\ten_{\bZ[\pi]}C_*(\tilde X).$$
\item[(b)] [Singular chains] Show that the action of $\pi$ on $\tilde{X}$ induces an action of
$\bZ[\pi]$ on the singular chain complex $C_*(\tilde{X})$ such that each $C_q(\tilde{X})$ is
a free $\bZ[\pi]$-module and $$C_*(X;A)\iso A\ten_{\bZ[\pi]}C_*(\tilde X).$$
\end{enumerate}
\item Let $\pi$ be a group and let $K(\pi,1)$ be a connected CW complex such that
$\pi_1(K(\pi,1))=\pi$ and $\pi_q(K(\pi,1))=0$ for $q\neq 1$. Use Problem 1 to show that
there is an isomorphism
$$H_*(K(\pi,1);A)\iso \Tor_*^{\bZ[\pi]}(A,\bZ).$$
\item Let $p: Y\rtarr X$ be a covering space with finite fibers, say of cardinality $n$.
Using singular chains, construct a homomorphism $t: H_*(X;A)\rtarr H_*(Y;A)$ such
that the composite $p_*\com t: H_*(X;A)\rtarr H_*(X;A)$ is multiplication by $n$;
$t$ is called a ``transfer\index{transfer homomorphism} homomorphism.''
\end{enumerate}
\chapter{Some more homological algebra}
\chapter{Axiomatic and cellular cohomology theory}
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}
\begin{enumerate}
\item Complete the proof of the uniqueness theorem for cohomology.
\end{enumerate}
In the following sequence of problems, we take cohomology with coefficients
in a commutative ring $R$ and we write $\ten$ for $\ten_R$.
\begin{enumerate}
\item[2.] Let $A$ and $B$ be subspaces of a space $X$. Construct a relative
cup product\index{cup product!relative}
$$H^p(X,A)\otimes H^q(X,B)\rtarr H^{p+q}(X,A\cup B)$$
and show that the following diagram is commutative:
$$\diagram
H^p(X,A)\otimes H^q(X,B) \rto \dto & H^{p+q}(X,A\cup B) \dto \\
H^p(X)\otimes H^q(X)\rto & H^{p+q}(X).\\
\enddiagram$$
The horizontal arrows are cup products; the vertical arrows are induced from
$X\rtarr (X,A)$, and so forth.
\item[3.] Let $X$ have a basepoint $*\in A\cap B$. Deduce a commutative diagram
$$\diagram
H^p(X,A)\otimes H^q(X,B) \rto \dto & H^{p+q}(X,A\cup B) \dto \\
\tilde{H}^p(X)\otimes \tilde{H}^q(X)\rto & \tilde{H}^{p+q}(X).\\
\enddiagram$$
\item[4.] Let $X = A\cup B$, where $A $ and $B$ are contractible and $A\cap B \neq\emptyset$.
Deduce that the cup product
$$\tilde{H}^p(X)\otimes \tilde{H}^q(X)\rtarr \tilde{H}^{p+q}(X)$$
is the zero homomorphism.
\item[5.] Let $X = \SI Y = Y\sma S^1$. Deduce that the cup product
$$\tilde{H}^p(X)\otimes \tilde{H}^q(X)\rtarr \tilde{H}^{p+q}(X)$$
is the zero homomorphism.
\end{enumerate}
Commentary: Additively, cohomology groups are ``stable,''\index{stable} in the sense that
$$\tilde{H}^p(Y) \iso \tilde{H}^{p+1}(\SI Y).$$
Cup products are ``unstable,'' in the sense that they vanish on suspensions.
This is an indication of how much more information they carry than the mere
additive groups. The proof given by this sequence of exercises actually
applies to any ``multiplicative'' cohomology theory,\index{cohomology theory!multiplicative}
that is, any theory that has suitable cup products.
\chapter{The Poincar\'e duality theorem}
\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}
\chapter{The index of manifolds; manifolds with boundary}
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}
\begin{enumerate}
\item For Abelian groups $\pi$ and $\rh$, show that $[K(\pi,n),K(\rh,n)]\iso \Hom(\pi,\rh)$.
(Hint: use the natural isomorphism $[X,K(\rh,n)]\iso \tilde{H}^n(X;\rh)$ and universal
coefficients.)
\item
\begin{enumerate}
\item[(a)] Let $f: \pi\rtarr \rh$ be a homomorphism of Abelian groups. Construct cohomology
operations $f^*: H^q(X;\pi)\rtarr H^q(X;\rh)$ for all $q$.
\item[(b)] Let $0\rtarr \pi\overto{f}\rh\overto{g}\si\rtarr 0$ be an exact sequence of
Abelian groups. Construct cohomology operations $\be: H^q(X;\si)\rtarr H^{q+1}(X;\pi)$ for
all $q$ such that the following is a long exact sequence:
$$ \cdots \rtarr H^q(X;\pi)\overto{f^*} H^q(X;\rh)\overto{g^*} H^q(X;\si)
\overto{\be} H^{q+1}(X;\pi)\rtarr \cdots.$$
The $\be$ are called Bockstein operations.\index{Bockstein operation}
\end{enumerate}
\item Using the calculation of $H^*(K(\bZ,2);\bZ_2)$ stated in the text, prove that
$Sq^1: H^q(X;\bZ_2)\rtarr H^{q+1}(X;\bZ_2)$ coincides with
the Bockstein operation associated to the short exact sequence
$0\rtarr \bZ_2 \rtarr \bZ_4 \rtarr \bZ_2 \rtarr 0$.
\item Prove that each $\PH^q$ of a stable cohomology operation $\sset{\PH^q}$ is a natural
homomorphism.
\item Write $H^*(\bR P^{\infty};\bZ_2)=\bZ_2[\al]$, $\deg \al=1$. Compute $Sq^i(\al^j)$
for all $i$ and $j$.
\end{enumerate}
\chapter{Characteristic classes of vector bundles}
\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}
\chapter{An introduction to cobordism}
\chapter*{Suggestions for further reading}
\end{document}