Basic Concepts of Algebraic Topology

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This text is intended as a one semester introduction to algebraic topology 
at the undergraduate and beginning graduate levels. Basically, it covers 
simplicial homology theory, the fundamental group, covering spaces, the 
higher homotopy groups and introductory singular homology theory. 
The text follows a broad historical outline and uses the proofs of the 
discoverers of the important theorems when this is consistent with the 
elementary level of the course. This method of presentation is intended to 
reduce the abstract nature of algebraic topology to a level that is palatable 
for the beginning student and to provide motivation and cohesion that are 
often lacking in abstact treatments. The text emphasizes the geometric 
approach to algebraic topology and attempts to show the importance of 
topological concepts by applying them to problems of geometry and 
analysis. 
The prerequisites for this course are calculus at the sophomore level, a 
one semester introduction to the theory of groups, a one semester introduction to point-set topology and some familiarity with vector spaces. Outlines 
of the prerequisite material can be found in the appendices at the end of 
the text. It is suggested that the reader not spend time initially working on 
the appendices, but rather that he read from the beginning of the text, 
referring to the appendices as his memory needs refreshing. The text is 
designed for use by college juniors of normal intelligence and does not 
require "mathematical maturity" beyond the junior level. 
The core of the course is the first four chapters—geometric complexes, 
simplicial homology groups, simplicial mappings, and the fundamental 
group. After completing Chapter 4, the reader may take the chapters in 
any order that suits him. Those particularly interested in the homology 
sequence and singular homology may choose, for example, to skip Chapter 
5 (covering spaces) and Chapter 6 (the higher homotopy groups) temporarily and proceed directly to Chapter 7.