Author | Rotman, Joseph J. author |
---|---|
Title | An Introduction to Algebraic Topology [electronic resource] / by Joseph J. Rotman |
Imprint | New York, NY : Springer New York, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4576-6 |
Descript | XIV, 438 p. online resource |
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 ?1 -- ?1(S1) -- 4 Singular Homology -- Holes and Greenโs 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 Kรผnneth 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 -- Notation