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**A Concise Course in Algebraic Topology** written by ** J. P. May **
This is an other great mathematics book cover the following topics.

What is algebraic topology?, The fundamental group, Dependence on the basepoint, Homotopy invariance, Calculations: p1(R) = 0 and p1(S1) = Z, The Brouwer fixed point theorem, The fundamental theorem of algebra

Categories, Functors, Natural transformations, Homotopy categories and homotopy equivalences, The fundamental groupoid, Limits and colimits, The van Kampen theorem, Examples of the van Kampen theorem

The definition of covering spaces, The unique path lifting property, Coverings of groupoids, Group actions and orbit categories, The classification of coverings of groupoids, The construction of coverings of groupoids, The classification of coverings of spaces, The construction of coverings of spaces

The definition of graphs, Edge paths and trees, The homotopy types of graphs, Covers of graphs and Euler characteristics, Applications to groups

The definition of compactly generated spaces, The category of compactly generated spaces

The definition of cofibrations, Mapping cylinders and cofibrations, Replacing maps by cofibrations, A criterion for a map to be a cofibration, Cofiber homotopy equivalence

The definition of fibrations, Path lifting functions and fibrations, Replacing maps by fibrations, A criterion for a map to be a fibration, Fiber homotopy equivalence, Change of fiber

Based homotopy classes of maps, Cones, suspensions, paths, loops, Based cofibrations, Cofiber sequences, Based fibrations, Fiber sequences, Connections between cofiber and fiber sequences

The definition of homotopy groups, Long exact sequences associated to pairs, Long exact sequences associated to fibrations, A few calculations, Change of basepoint, n-Equivalences, weak equivalences, and a technical lemma

The definition and some examples of CW complexes, ome constructions on CW complexes, HELP and the Whitehead theorem, The cellular approximation theorem, Approximation of spaces by CW complexes, Approximation of pairs by CW pairs, Approximation of excisive triads by CW triads

Statement of the homotopy excision theorem, The Freudenthal suspension theorem, Proof of the homotopy excision theorem

Chain complexes, Maps and homotopies of maps of chain complexes, Tensor products of chain complexes, Short and long exact sequences

Axioms for homology, Cellular homology, Verification of the axioms, The cellular chains of products, Some examples: T , K, and RPn

Reduced homology; based versus unbased spaces, Cofibrations and the homology of pairs, Suspension and the long exact sequence of pairs, Axioms for reduced homology, Mayer-Vietoris sequences, The homology of colimits

The Hurewicz theorem, The uniqueness of the homology of CW complexes

The singular chain complex, Geometric realization, Proofs of the theorems, Simplicial objects in algebraic topology, Classifying spaces and K(p, n)s

Universal coefficients in homology, The K¨unneth theorem, Hom functors and universal coefficients in cohomology, Proof of the universal coefficient theorem, Relations between ? and Hom

Axioms for cohomology, Cellular and singular cohomology, Cup products in cohomology, An example: RPn and the Borsuk-Ulam theorem, Obstruction theory

Reduced cohomology groups and their properties, Axioms for reduced cohomology, Mayer-Vietoris sequences in cohomology, Lim1 and the cohomology of colimits, The uniqueness of the cohomology of CW complexes

Statement of the theorem, The definition of the cap product, Orientations and fundamental classes, The proof of the vanishing theorem, The proof of the Poincar´e duality theorem, The orientation cover

The Euler characteristic of compact manifolds, The index of compact oriented manifolds, Manifolds with boundary, Poincar´e duality for manifolds with boundary, The index of manifolds that are boundaries

K(p, n)s and homology, K(p, n)s and cohomology, Cup and cap products, Postnikov systems, Cohomology operations

The classification of vector bundles, Characteristic classes for vector bundles, Stiefel-Whitney classes of manifolds, Characteristic numbers of manifolds, Thom spaces and the Thom isomorphism theorem, The construction of the Stiefel-Whitney classes, Chern, Pontryagin, and Euler classes, A glimpse at the general theory

The definition of K-theory, The Bott periodicity theorem, The splitting principle and the Thom isomorphism, The Chern character; almost complex structures on spheres, The Adams operations, The Hopf invariant one problem and its applications

The cobordism groups of smooth closed manifolds, Sketch proof that N* is isomorphic to p*(TO), Prespectra and the algebra H*(TO; Z2), The Steenrod algebra and its coaction on H*(TO), The relationship to Stiefel-Whitney numbers, Spectra and the computation of p*(TO) = p*(MO), An introduction to the stable category

A classic book and historical references, Textbooks in algebraic topology and homotopy theory, Books on CW complexes, Differential forms and Morse theory, Equivariant algebraic topology, Category theory and homological algebra, Simplicial sets in algebraic topology, The Serre spectral sequence and Serre class theory, The Eilenberg-Moore spectral sequence, Cohomology operations, Vector bundles, Characteristic classes, K-theory, Hopf algebras; the Steenrod algebra, Adams spectral sequence, Cobordism, Generalized homology theory and stable homotopy theory, Quillen model categories, Localization and completion; rational homotopy theory, Infinite loop space theory, Complex cobordism and stable homotopy theory, Follow-ups to this book

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