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Algebraic Topology by Allen Hatcher


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Algebraic Topology written by Allen Hatcher This is an other great mathematics book 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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