\contentsline {chapter}{\tocchapter {Chapter}{}{Introduction}}{1}
\contentsline {chapter}{\tocchapter {Chapter}{1}{The fundamental group and some of its applications}}{5}
\contentsline {section}{\tocsection {}{1}{What is algebraic topology?}}{5}
\contentsline {section}{\tocsection {}{2}{The fundamental group}}{6}
\contentsline {section}{\tocsection {}{3}{Dependence on the basepoint}}{7}
\contentsline {section}{\tocsection {}{4}{Homotopy invariance}}{7}
\contentsline {section}{\tocsection {}{5}{Calculations: $\pi _1(\@mathbb {R})=0$ and $\pi _1(S^1)=\@mathbb {Z}$}}{8}
\contentsline {section}{\tocsection {}{6}{The Brouwer fixed point theorem}}{10}
\contentsline {section}{\tocsection {}{7}{The fundamental theorem of algebra}}{10}
\contentsline {chapter}{\tocchapter {Chapter}{2}{Categorical language and the van Kampen theorem}}{13}
\contentsline {section}{\tocsection {}{1}{Categories}}{13}
\contentsline {section}{\tocsection {}{2}{Functors}}{13}
\contentsline {section}{\tocsection {}{3}{Natural transformations}}{14}
\contentsline {section}{\tocsection {}{4}{Homotopy categories and homotopy equivalences}}{14}
\contentsline {section}{\tocsection {}{5}{The fundamental groupoid}}{15}
\contentsline {section}{\tocsection {}{6}{Limits and colimits}}{16}
\contentsline {section}{\tocsection {}{7}{The van Kampen theorem}}{17}
\contentsline {section}{\tocsection {}{8}{Examples of the van Kampen theorem}}{19}
\contentsline {chapter}{\tocchapter {Chapter}{3}{Covering spaces}}{21}
\contentsline {section}{\tocsection {}{1}{The definition of covering spaces}}{21}
\contentsline {section}{\tocsection {}{2}{The unique path lifting property}}{22}
\contentsline {section}{\tocsection {}{3}{Coverings of groupoids}}{22}
\contentsline {section}{\tocsection {}{4}{Group actions and orbit categories}}{24}
\contentsline {section}{\tocsection {}{5}{The classification of coverings of groupoids}}{25}
\contentsline {section}{\tocsection {}{6}{The construction of coverings of groupoids}}{27}
\contentsline {section}{\tocsection {}{7}{The classification of coverings of spaces}}{28}
\contentsline {section}{\tocsection {}{8}{The construction of coverings of spaces}}{30}
\contentsline {chapter}{\tocchapter {Chapter}{4}{Graphs}}{35}
\contentsline {section}{\tocsection {}{1}{The definition of graphs}}{35}
\contentsline {section}{\tocsection {}{2}{Edge paths and trees}}{35}
\contentsline {section}{\tocsection {}{3}{The homotopy types of graphs}}{36}
\contentsline {section}{\tocsection {}{4}{Covers of graphs and Euler characteristics}}{37}
\contentsline {section}{\tocsection {}{5}{Applications to groups}}{37}
\contentsline {chapter}{\tocchapter {Chapter}{5}{Compactly generated spaces}}{39}
\contentsline {section}{\tocsection {}{1}{The definition of compactly generated spaces}}{39}
\contentsline {section}{\tocsection {}{2}{The category of compactly generated spaces}}{40}
\contentsline {chapter}{\tocchapter {Chapter}{6}{Cofibrations}}{43}
\contentsline {section}{\tocsection {}{1}{The definition of cofibrations}}{43}
\contentsline {section}{\tocsection {}{2}{Mapping cylinders and cofibrations}}{44}
\contentsline {section}{\tocsection {}{3}{Replacing maps by cofibrations}}{45}
\contentsline {section}{\tocsection {}{4}{A criterion for a map to be a cofibration}}{45}
\contentsline {section}{\tocsection {}{5}{Cofiber homotopy equivalence}}{46}
\contentsline {chapter}{\tocchapter {Chapter}{7}{Fibrations}}{49}
\contentsline {section}{\tocsection {}{1}{The definition of fibrations}}{49}
\contentsline {section}{\tocsection {}{2}{Path lifting functions and fibrations}}{49}
\contentsline {section}{\tocsection {}{3}{Replacing maps by fibrations}}{50}
\contentsline {section}{\tocsection {}{4}{A criterion for a map to be a fibration}}{51}
\contentsline {section}{\tocsection {}{5}{Fiber homotopy equivalence}}{52}
\contentsline {section}{\tocsection {}{6}{Change of fiber}}{53}
\contentsline {chapter}{\tocchapter {Chapter}{8}{Based cofiber and fiber sequences}}{57}
\contentsline {section}{\tocsection {}{1}{Based homotopy classes of maps}}{57}
\contentsline {section}{\tocsection {}{2}{Cones, suspensions, paths, loops}}{57}
\contentsline {section}{\tocsection {}{3}{Based cofibrations}}{58}
\contentsline {section}{\tocsection {}{4}{Cofiber sequences}}{59}
\contentsline {section}{\tocsection {}{5}{Based fibrations}}{61}
\contentsline {section}{\tocsection {}{6}{Fiber sequences}}{61}
\contentsline {section}{\tocsection {}{7}{Connections between cofiber and fiber sequences}}{63}
\contentsline {chapter}{\tocchapter {Chapter}{9}{Higher homotopy groups}}{65}
\contentsline {section}{\tocsection {}{1}{The definition of homotopy groups}}{65}
\contentsline {section}{\tocsection {}{2}{Long exact sequences associated to pairs}}{65}
\contentsline {section}{\tocsection {}{3}{Long exact sequences associated to fibrations}}{66}
\contentsline {section}{\tocsection {}{4}{A few calculations}}{66}
\contentsline {section}{\tocsection {}{5}{Change of basepoint}}{68}
\contentsline {section}{\tocsection {}{6}{$n$-Equivalences, weak equivalences, and a technical lemma}}{69}
\contentsline {chapter}{\tocchapter {Chapter}{10}{CW complexes}}{73}
\contentsline {section}{\tocsection {}{1}{The definition and some examples of CW complexes}}{73}
\contentsline {section}{\tocsection {}{2}{Some constructions on CW complexes}}{74}
\contentsline {section}{\tocsection {}{3}{HELP and the Whitehead theorem}}{75}
\contentsline {section}{\tocsection {}{4}{The cellular approximation theorem}}{76}
\contentsline {section}{\tocsection {}{5}{Approximation of spaces by CW complexes}}{77}
\contentsline {section}{\tocsection {}{6}{Approximation of pairs by CW pairs}}{78}
\contentsline {section}{\tocsection {}{7}{Approximation of excisive triads by CW triads}}{79}
\contentsline {chapter}{\tocchapter {Chapter}{11}{The homotopy excision and suspension theorems}}{83}
\contentsline {section}{\tocsection {}{1}{Statement of the homotopy excision theorem}}{83}
\contentsline {section}{\tocsection {}{2}{The Freudenthal suspension theorem}}{85}
\contentsline {section}{\tocsection {}{3}{Proof of the homotopy excision theorem}}{86}
\contentsline {chapter}{\tocchapter {Chapter}{12}{A little homological algebra}}{91}
\contentsline {section}{\tocsection {}{1}{Chain complexes}}{91}
\contentsline {section}{\tocsection {}{2}{Maps and homotopies of maps of chain complexes}}{91}
\contentsline {section}{\tocsection {}{3}{Tensor products of chain complexes}}{92}
\contentsline {section}{\tocsection {}{4}{Short and long exact sequences}}{93}
\contentsline {chapter}{\tocchapter {Chapter}{13}{Axiomatic and cellular homology theory}}{95}
\contentsline {section}{\tocsection {}{1}{Axioms for homology}}{95}
\contentsline {section}{\tocsection {}{2}{Cellular homology}}{97}
\contentsline {section}{\tocsection {}{3}{Verification of the axioms}}{100}
\contentsline {section}{\tocsection {}{4}{The cellular chains of products}}{101}
\contentsline {section}{\tocsection {}{5}{Some examples: $T$, $K$, and $\@mathbb {R}P^n$}}{103}
\contentsline {chapter}{\tocchapter {Chapter}{14}{Derivations of properties from the axioms}}{107}
\contentsline {section}{\tocsection {}{1}{Reduced homology; based versus unbased spaces}}{107}
\contentsline {section}{\tocsection {}{2}{Cofibrations and the homology of pairs}}{108}
\contentsline {section}{\tocsection {}{3}{Suspension and the long exact sequence of pairs}}{109}
\contentsline {section}{\tocsection {}{4}{Axioms for reduced homology}}{110}
\contentsline {section}{\tocsection {}{5}{Mayer-Vietoris sequences}}{112}
\contentsline {section}{\tocsection {}{6}{The homology of colimits}}{114}
\contentsline {chapter}{\tocchapter {Chapter}{15}{The Hurewicz and uniqueness theorems}}{117}
\contentsline {section}{\tocsection {}{1}{The Hurewicz theorem}}{117}
\contentsline {section}{\tocsection {}{2}{The uniqueness of the homology of CW complexes}}{119}
\contentsline {chapter}{\tocchapter {Chapter}{16}{Singular homology theory}}{123}
\contentsline {section}{\tocsection {}{1}{The singular chain complex}}{123}
\contentsline {section}{\tocsection {}{2}{Geometric realization}}{124}
\contentsline {section}{\tocsection {}{3}{Proofs of the theorems}}{125}
\contentsline {section}{\tocsection {}{4}{Simplicial objects in algebraic topology}}{126}
\contentsline {section}{\tocsection {}{5}{Classifying spaces and $K(\pi ,n)$s}}{128}
\contentsline {chapter}{\tocchapter {Chapter}{17}{Some more homological algebra}}{131}
\contentsline {section}{\tocsection {}{1}{Universal coefficients in homology}}{131}
\contentsline {section}{\tocsection {}{2}{The K\"{u}nneth theorem}}{132}
\contentsline {section}{\tocsection {}{3}{Hom functors and universal coefficients in cohomology}}{133}
\contentsline {section}{\tocsection {}{4}{Proof of the universal coefficient theorem}}{135}
\contentsline {section}{\tocsection {}{5}{Relations between $\otimes $ and Hom}}{136}
\contentsline {chapter}{\tocchapter {Chapter}{18}{Axiomatic and cellular cohomology theory}}{137}
\contentsline {section}{\tocsection {}{1}{Axioms for cohomology}}{137}
\contentsline {section}{\tocsection {}{2}{Cellular and singular cohomology}}{138}
\contentsline {section}{\tocsection {}{3}{Cup products in cohomology}}{139}
\contentsline {section}{\tocsection {}{4}{An example: $\@mathbb {R}P^n$ and the Borsuk-Ulam theorem}}{140}
\contentsline {section}{\tocsection {}{5}{Obstruction theory}}{142}
\contentsline {chapter}{\tocchapter {Chapter}{19}{Derivations of properties from the axioms}}{145}
\contentsline {section}{\tocsection {}{1}{Reduced cohomology groups and their properties}}{145}
\contentsline {section}{\tocsection {}{2}{Axioms for reduced cohomology}}{146}
\contentsline {section}{\tocsection {}{3}{Mayer-Vietoris sequences in cohomology}}{147}
\contentsline {section}{\tocsection {}{4}{Lim$^1$ and the cohomology of colimits}}{148}
\contentsline {section}{\tocsection {}{5}{The uniqueness of the cohomology of CW complexes}}{149}
\contentsline {chapter}{\tocchapter {Chapter}{20}{The Poincar\'e duality theorem}}{151}
\contentsline {section}{\tocsection {}{1}{Statement of the theorem}}{151}
\contentsline {section}{\tocsection {}{2}{The definition of the cap product}}{153}
\contentsline {section}{\tocsection {}{3}{Orientations and fundamental classes}}{155}
\contentsline {section}{\tocsection {}{4}{The proof of the vanishing theorem}}{158}
\contentsline {section}{\tocsection {}{5}{The proof of the Poincar\'e duality theorem}}{160}
\contentsline {section}{\tocsection {}{6}{The orientation cover}}{163}
\contentsline {chapter}{\tocchapter {Chapter}{21}{The index of manifolds; manifolds with boundary}}{165}
\contentsline {section}{\tocsection {}{1}{The Euler characteristic of compact manifolds}}{165}
\contentsline {section}{\tocsection {}{2}{The index of compact oriented manifolds}}{166}
\contentsline {section}{\tocsection {}{3}{Manifolds with boundary}}{168}
\contentsline {section}{\tocsection {}{4}{Poincar\'e duality for manifolds with boundary}}{169}
\contentsline {section}{\tocsection {}{5}{The index of manifolds that are boundaries}}{171}
\contentsline {chapter}{\tocchapter {Chapter}{22}{Homology, cohomology, and $K(\pi ,n)$s}}{175}
\contentsline {section}{\tocsection {}{1}{$K(\pi ,n)$s and homology}}{175}
\contentsline {section}{\tocsection {}{2}{$K(\pi ,n)$s and cohomology}}{177}
\contentsline {section}{\tocsection {}{3}{Cup and cap products}}{179}
\contentsline {section}{\tocsection {}{4}{Postnikov systems}}{182}
\contentsline {section}{\tocsection {}{5}{Cohomology operations}}{184}
\contentsline {chapter}{\tocchapter {Chapter}{23}{Characteristic classes of vector bundles}}{187}
\contentsline {section}{\tocsection {}{1}{The classification of vector bundles}}{187}
\contentsline {section}{\tocsection {}{2}{Characteristic classes for vector bundles}}{189}
\contentsline {section}{\tocsection {}{3}{Stiefel-Whitney classes of manifolds}}{191}
\contentsline {section}{\tocsection {}{4}{Characteristic numbers of manifolds}}{193}
\contentsline {section}{\tocsection {}{5}{Thom spaces and the Thom isomorphism theorem}}{194}
\contentsline {section}{\tocsection {}{6}{The construction of the Stiefel-Whitney classes}}{196}
\contentsline {section}{\tocsection {}{7}{Chern, Pontryagin, and Euler classes}}{197}
\contentsline {section}{\tocsection {}{8}{A glimpse at the general theory}}{200}
\contentsline {chapter}{\tocchapter {Chapter}{24}{An introduction to $K$-theory}}{203}
\contentsline {section}{\tocsection {}{1}{The definition of $K$-theory}}{203}
\contentsline {section}{\tocsection {}{2}{The Bott periodicity theorem}}{206}
\contentsline {section}{\tocsection {}{3}{The splitting principle and the Thom isomorphism}}{208}
\contentsline {section}{\tocsection {}{4}{The Chern character; almost complex structures on spheres}}{211}
\contentsline {section}{\tocsection {}{5}{The Adams operations}}{213}
\contentsline {section}{\tocsection {}{6}{The Hopf invariant one problem and its applications}}{215}
\contentsline {chapter}{\tocchapter {Chapter}{25}{An introduction to cobordism}}{219}
\contentsline {section}{\tocsection {}{1}{The cobordism groups of smooth closed manifolds}}{219}
\contentsline {section}{\tocsection {}{2}{Sketch proof that $\@scr {N}_*$ is isomorphic to $\pi _*(TO)$}}{220}
\contentsline {section}{\tocsection {}{3}{Prespectra and the algebra $H_*(TO;\@mathbb {Z}_2)$}}{223}
\contentsline {section}{\tocsection {}{4}{The Steenrod algebra and its coaction on $H_*(TO)$}}{226}
\contentsline {section}{\tocsection {}{5}{The relationship to Stiefel-Whitney numbers}}{228}
\contentsline {section}{\tocsection {}{6}{Spectra and the computation of $\pi _*(TO) =\pi _*(MO)$}}{230}
\contentsline {section}{\tocsection {}{7}{An introduction to the stable category}}{232}
\contentsline {chapter}{\tocchapter {Chapter}{}{Suggestions for further reading}}{235}
\contentsline {section}{\tocsection {}{1}{A classic book and historical references}}{235}
\contentsline {section}{\tocsection {}{2}{Textbooks in algebraic topology and homotopy theory}}{235}
\contentsline {section}{\tocsection {}{3}{Books on CW complexes}}{236}
\contentsline {section}{\tocsection {}{4}{Differential forms and Morse theory}}{236}
\contentsline {section}{\tocsection {}{5}{Equivariant algebraic topology}}{237}
\contentsline {section}{\tocsection {}{6}{Category theory and homological algebra}}{237}
\contentsline {section}{\tocsection {}{7}{Simplicial sets in algebraic topology}}{237}
\contentsline {section}{\tocsection {}{8}{The Serre spectral sequence and Serre class theory}}{237}
\contentsline {section}{\tocsection {}{9}{The Eilenberg-Moore spectral sequence}}{237}
\contentsline {section}{\tocsection {}{10}{Cohomology operations}}{238}
\contentsline {section}{\tocsection {}{11}{Vector bundles}}{238}
\contentsline {section}{\tocsection {}{12}{Characteristic classes}}{238}
\contentsline {section}{\tocsection {}{13}{$K$-theory}}{239}
\contentsline {section}{\tocsection {}{14}{Hopf algebras; the Steenrod algebra, Adams spectral sequence}}{239}
\contentsline {section}{\tocsection {}{15}{Cobordism}}{240}
\contentsline {section}{\tocsection {}{16}{Generalized homology theory and stable homotopy theory}}{240}
\contentsline {section}{\tocsection {}{17}{Quillen model categories}}{240}
\contentsline {section}{\tocsection {}{18}{Localization and completion; rational homotopy theory}}{241}
\contentsline {section}{\tocsection {}{19}{Infinite loop space theory}}{241}
\contentsline {section}{\tocsection {}{20}{Complex cobordism and stable homotopy theory}}{242}
\contentsline {section}{\tocsection {}{21}{Follow-ups to this book}}{242}