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Algebraic Topology by Allen Hatcher
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**About this book :- **
**Algebraic Topology ** written by
** Allen Hatcher **

This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.

**Book Detail :- **
** Title: ** Algebraic Topology
** Edition: **
** Author(s): ** Allen Hatcher
** Publisher: ** Cambridge University Press
** Series: **
** Year: ** 2001
** Pages: ** 560
** Type: ** PDF
** Language: ** English
** ISBN: ** 0521795400,9780521795401
** Country: ** US

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**Book Contents :- **
**Algebraic Topology ** written by
** Allen Hatcher **
cover the following topics.
**0. Some Underlying Geometric Notions**

Homotopy and Homotopy Type 1. Cell Complexes, Operations on Spaces, Two Criteria for Homotopy Equivalence, The Homotopy Extension Property
**The Fundamental Group**

Basic Constructions, Paths and Homotopy, The Fundamental Group of the Circle, Induced Homomorphisms, Van Kampen’s Theorem, Free Products of Groups, The van Kampen Theorem, Applications to Cell Complexes, Covering Spaces, Lifting Properties, The Classification of Covering Spaces, Deck Transformations and Group Actions, Graphs and Free Groups, K(G,1) Spaces and Graphs of Groups
**2. Homology **

Simplicial and Singular Homology, ∆ Complexes, Simplicial Homology, Singular Homology, Homotopy Invariance, Exact Sequences and Excision, The Equivalence of Simplicial and Singular Homology, Computations and Applications, Degree, Cellular Homology, Mayer-Vietoris Sequences, Homology with Coefficients, The Formal Viewpoint, Axioms for Homology, Categories and Functors, Additional Topics, Homology and Fundamental Group, Classical Applications, Simplicial Approximation
**3. Cohomology**

Cohomology Groups, The Universal Coefficient Theorem 190. Cohomology of Spaces, Cup Product, The Cohomology Ring, A Kunneth Formula, Spaces with Polynomial Cohomology, Poincar´e Duality, Orientations and Homology, The Duality Theorem, Connection with Cup Product, Other Forms of Duality, Additional Topics, Universal Coefficients for Homology, The General Kunneth Formula, H–Spaces and Hopf Algebras, The Cohomology of SO(n), Bockstein Homomorphisms, Limits and Ext, Transfer Homomorphisms, Local Coefficients
**4. Homotopy Theory**

Homotopy Groups, Definitions and Basic Constructions, Whitehead’s Theorem, Cellular Approximation 348. CW Approximation, Elementary Methods of Calculation, Excision for Homotopy Groups, The Hurewicz Theorem, Fiber Bundles, Stable Homotopy Groups, Connections with Cohomology, The Homotopy Construction of Cohomology, Fibrations, Postnikov Towers, Obstruction Theory, Basepoints and Homotopy, The Hopf Invariant, Minimal Cell Structures, Cohomology of Fiber Bundles, The Brown Representability Theorem, Spectra and Homology Theories, Gluing Constructions, Eckmann-Hilton Duality, Stable Splittings of Spaces, The Loopspace of a Suspension, The Dold-Thom Theorem, Steenrod Squares and Powers
**Appendix**

Topology of Cell Complexes, The Compact-Open Topology, The Homotopy Extension Property, Simplicial CW Structures

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