Differential Topology First Steps

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ISBN: 9780486453170 | 0486453170
Cover: Paperback
Copyright: 10/27/2006

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Keeping mathematical prerequisites to a minimum, this undergraduate-level text stimulates students' intuitive understanding of topology while avoiding the more difficult subtleties and technicalities. Its focus is the method of spherical modifications and the study of critical points of functions on manifolds.

Editors' Foreword	p. v
Preface	p. vii
Set Theoretic Symbols	p. xiii
Topological Spaces	p. 1
Neighborhoods	p. 1
Open and Closed Sets	p. 4
Continuous Maps	p. 6
Topological Products	p. 7
Connectedness	p. 8
Compactness	p. 11
Differentiable Manifolds	p. 14
Introduction	p. 14
Differentiable Functions and Maps	p. 18
Differentiable Manifolds	p. 20
Local Coordinates and Differentiable Functions	p. 22
Differentiable Maps	p. 26
Rank of a Differentiable Map	p. 29
Manifolds with Boundary	p. 30
Submanifolds	p. 33
The Definition	p. 33
Manifolds in Euclidean Space	p. 35
The Embedding Theorem	p. 41
Embedding a Manifold with Boundary	p. 43
Tangent Spaces and Critical Points	p. 46
Tangent Lines	p. 46
Critical Points	p. 48
Nondegenerate Critical Points	p. 54
A Stronger Embedding Theorem	p. 58
Critical and Noncritical Levels	p. 61
Definitions and Examples	p. 61
A Neighborhood of a Critical Level; an Example	p. 65
Neighborhood of a Critical Level; General Discussion	p. 67
Neighborhood of a Critical Point	p. 69
Neighborhood of a Critical Level; Summing Up	p. 74
Spherical Modifications	p. 76
Introduction	p. 76
Direct Embedding	p. 77
Definition of Modifications	p. 80
The Trace of a Modification	p. 84
Cobounding Manifolds	p. 88
Displacement and Isotopy	p. 89
General Position	p. 93
Rearrangement of Modifications	p. 95
An Application to 3-Manifolds	p. 97
Interpretation of Theorem 6-5 in Terms of Critical Points	p. 98
Two-Dimensional Manifolds	p. 99
Introduction	p. 99
Orientable 2-Manifolds	p. 100
The Nonorientable Case	p. 111
Second Steps	p. 118
Killing of Homotopy Classes	p. 118
Complementary Modifications and Cancellation	p. 121
A Theorem on 3-Manifolds	p. 125
Bibliography	p. 127
Index	p. 129
Table of Contents provided by Ingram. All Rights Reserved.

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Differential Topology First Steps

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