title: Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics) (v. 145)
author:  James W. Vick 
description: This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincare duality, and fixed point theory. Throughout the book, the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. This second edition has been expanded to include a new chapter on covering spaces, as well as additional illuminating exercises. The conceptual approach is again used to show how lifting problems give rise to the fundamental group and its properties.
zlib_text: A Concise Course in Algebraic Topology J. P. May
  
  Contents
  
   **Introduction**
  
   **1**
  
  Chapter 1.
  
  The fundamental group and some of its applications 5
  
  1\.
  
  What is algebraic topology?
  
  5
  
  2\.
  
  The fundamental group
  
  6
  
  3\.
  
  Dependence on the basepoint
  
  7
  
  4\.
  
  Homotopy invariance
  
  7
  
  5\.
  
  Calculations: π1(R) = 0 and π1(S1) = Z
  
  8
  
  6\.
  
  The Brouwer fixed point theorem
  
  10
  
  7\.
  
  The fundamental theorem of algebra
  
  10
  
  Chapter 2.
  
  Categorical language and the van Kampen theorem 13
  
  1\.
  
  Categories
  
  13
  
  2\.
  
  Functors
  
  13
  
  3\.
  
  Natural transformations
  
  14
  
  4\.
  
  Homotopy categories and homotopy equivalences 14
  
  5\.
  
  The fundamental groupoid
  
  15
  
  6\.
  
  Limits and colimits
  
  16
  
  7\.
  
  The van Kampen theorem
  
  17
  
  8\.
  
  Examples of the van Kampen theorem
  
  19
  
  Chapter 3.
  
  Covering spaces
  
  21
  
  1\.
  
  The definition of covering spaces
  
  21
  
  2\.
  
  The unique path lifting property
  
  22
  
  3\.
  
  Coverings of groupoids
  
  22
  
  4\.
  
  Group actions and orbit categories
  
  24
  
  5\.
  
  The classification of coverings of grou