An Introduction to Algebraic Topology

Description

Cover Title Page Preface To the Reader 0 Introduction Notation Brouwer Fixed Point Theorem Categories and Functors 1 Some Basic Topological Notions Homotopy Convexity, Contractibility, and Cones Paths and Path Connectedness 2 Simplexes Affine Spaces Affine Maps 3 The Fundamental Group The Fundamental Groupoid The Functor pi1 Pi1(S1) 4 Singular Homology Holes and Green's Theorem Free Abelian Groups The Singular Complex and Homology Functors Dimension Axiom and Compact Supports The Homotopy Axiom The Hurewicz Theorem 5 Long Exact Sequences The Category Comp Exact Homology Sequences Reduced Homology 6 Excision and Applications Excision and Mayer-Vietoris Homology of Spheres and Some Applications Barycentric Subdivision and the Proof of Excision More Applications to Euclidean Space 7 Simplicial Complexes Definitions Simplicial Approximation Abstract Simplicial Complexes Simplicial Homology Comparison with Singular Homology Calculations Fundamental Groups of Polyhedra The Seifert-van Kampen Theorem 8 CW Complexes Hausdorff Quotient Spaces Attaching Cells Homology and Attaching Cells CW Complexes Cellular Homology 9 Natural Transformations Definitions and Examples Eilenberg-Steenrod Axioms Chain Equivalences Acyclic Models Lefschetz Fixed Point Theorem Tensor Products Universal Coefficients Eilenberg-Zilber Theorem and the K�nneth Formula 10 Covering Spaces Basic Properties Covering Transformations Existence Orbit Spaces 11 Homotopy Groups Function Spaces Group Objects and Cogroup Objects Loop Space and Suspension Homotopy Groups Exact Sequences Fibrations A Glimpse Ahead 12 Cohomology Differential Forms Cohomology Groups Universal Coefficients Theorems for Cohomology Cohomology Rings Computations and Applications Bibliography Notation Index