Basic Concepts of Algebraic Topology

1. Front Matter

   Pages i-x

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2. Geometric Complexes and Polyhedra

   • Fred H. Croom

   Pages 1-15

3. Simplicial Homology Groups

   • Fred H. Croom

   Pages 16-38

4. Simplicial Approximation

   • Fred H. Croom

   Pages 39-59

5. The Fundamental Group

   • Fred H. Croom

   Pages 60-82

6. Covering Spaces

   • Fred H. Croom

   Pages 83-104

7. The Higher Homotopy Groups

   • Fred H. Croom

   Pages 105-127

8. Further Developments in Homology

   • Fred H. Croom

   Pages 128-153

9. Back Matter

   Pages 155-180

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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 introduc­ tion 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.

• Algebra
• Basic
• Derivation
• Manifold
• Morphism
• Topology
• theorem

• The University of the South, Sewanee, USA

  Fred H. Croom

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