Synopses & Reviews
A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.
Review
"Delightful reading that catches the interplay of geometry and algebra. There is an unusually large number of instructive exercises that, together with the text, cover the range from historical foundations to the categorical tools of homological algebra." AMERICAN MATHEMATICAL MONTHLY
Synopsis
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."
Synopsis
Handsome text provides graduate students with an exceptionally accessible introduction to the ideas and methods of algebraic topology. The author writes with graceful lucidity, and gives careful attention to motivational matters. Exercises punctuate the twelve chapters at frequent intervals. A valua
Synopsis
A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.
Synopsis
This book offers a detailed exposition, with exercises, of the basic ideas of algebraic topology: homology, homotopy groups, and cohomology rings. Avoiding excessive generality, the author explains the origins of abstract concepts as they are introduced.
Table of Contents
Preface. 0: Introduction. 1: Some Basic Topological Notions. 2: Simplexes. 3: The Fundamental Group. 4: Singular Homology. 5: Long Exact Sequences. 6: Excision and Applications. 7: Simplicial Complexes. 8: CW Complexes. 9: Natural Transformations. 10: Covering Spaces. 11: Homotopy Groups. 12: Cohomology.