Overview

There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini­ tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi­ cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim­ plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces.

Full Product Details

Author:   Joseph J. Rotman
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   1st ed. 1988. Corr. 4th printing 1998
Volume:   119
Dimensions:   Width: 15.50cm , Height: 2.50cm , Length: 23.50cm
Weight:   1.810kg
ISBN:  

9780387966786

ISBN 10:   0387966781
Pages:   437
Publication Date:   17 August 1988
Audience:   College/higher education ,  Undergraduate
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
We will order this item for you from a manufatured on demand supplier.

Table of Contents

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 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 Künneth 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.

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