A Concise Course in Algebraic Topology by J. P. May

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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.

The fundamental group and some of its applications
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

Categorical language and the van Kampen theorem
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

Covering spaces
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

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

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

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

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

Higher homotopy groups
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

CW complexes
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

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

A little homological algebra
Chain complexes, Maps and homotopies of maps of chain complexes, Tensor products of chain complexes, Short and long exact sequences

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

Derivations of properties from the axioms
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 and uniqueness theorems
The Hurewicz theorem, The uniqueness of the homology of CW complexes

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

Some more homological algebra
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

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

Derivations of properties from the axioms
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

The Poincar´e duality theorem
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 index of manifolds; manifolds with boundary
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

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

Characteristic classes of vector bundles
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

An introduction to K-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

An introduction to cobordism
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

Suggestions for further reading
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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