\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}