An Introduction to Algebraic Topology

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ISBN: 9780486457864 | 0486457869
Cover: Paperback
Copyright: 2/27/2007

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This self-contained treatment assumes only some knowledge of real numbers and real analysis. The first three chapters focus on the basics of point-set topology, after which the text proceeds to homology groups and continuous mapping, barycentric subdivision, and simplicial complexes. Exercises form an integral part of the text. 1961 edition.

Prerequisites
Set theory	p. 1
Algebra	p. 3
Euclidean spaces	p. 6
Introduction
Continuity and neighbourhoods	p. 7
The abstract concept of neighbourhood	p. 8
Topological Spaces
Definition of a topological space	p. 14
Open sets	p. 17
Another definition of a topological space	p. 21
Subspaces of a given space	p. 26
Limits	p. 29
Limit points	p. 33
Closure of a set	p. 36
Frontier of a set	p. 40
Topological Properties of Spaces
Continuous mappings and homeomorphisms	p. 42
Compact spaces	p. 50
Arcwise connected spaces	p. 55
Connected spaces	p. 57
The Fundamental Group
Homotopy	p. 63
Homotopy classes	p. 69
The fundamental group	p. 74
Change of base-point	p. 85
Topological invariance	p. 89
The Homology Groups
Geometrical motivation for homology theory	p. 92
Euclidean simplexes	p. 95
Linear mappings	p. 100
Singular simplexes on a space	p. 103
Chains on a space	p. 104
The boundary of a simplex	p. 105
Boundaries and cycles on any space	p. 107
Homologous cycles and homology groups	p. 111
Relative homology	p. 113
Continuous Mappings and the Homology Groups
The induced homomorphism	p. 117
Topological invariance of the homology groups	p. 119
Homotopic mappings and the homology groups	p. 120
Prisms	p. 123
Homotopic mappings and the homology groups (contd.)	p. 128
Barycentric Subdivision and Excision
Motivation for barycentric subdivision	p. 133
The operator B	p. 137
The operator H	p. 140
Reduction to small simplexes	p. 142
The excision theorem	p. 146
The Homology Sequence
The exact sequence	p. 150
Homology groups in some special cases	p. 153
Homology groups in cells and spheres	p. 162
Simplicial Complexes
Definition of complexes	p. 168
The direct sum theorem for complexes	p. 171
A generator for H[subscript r](S,S)	p. 175
The homology groups of a simplicial complex	p. 180
Oriented chains and cycles	p. 184
The oriented boundary operator	p. 188
Guide to Further Reading	p. 195
Index	p. 197
Table of Contents provided by Ingram. All Rights Reserved.

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An Introduction to Algebraic Topology

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