Differential Topology First Steps

ISBN-10: 0486453170

ISBN-13: 9780486453170

Edition: 2006

Authors: Andrew H. Wallace

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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.
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Book details

Copyright year: 2006
Publisher: Dover Publications, Incorporated
Publication date: 10/27/2006
Binding: Paperback
Pages: 144
Size: 5.50" wide x 8.50" long x 0.50" tall
Weight: 0.374
Language: English

Editors' Foreword
Set Theoretic Symbols
Topological Spaces
Open and Closed Sets
Continuous Maps
Topological Products
Differentiable Manifolds
Differentiable Functions and Maps
Differentiable Manifolds
Local Coordinates and Differentiable Functions
Differentiable Maps
Rank of a Differentiable Map
Manifolds with Boundary
The Definition
Manifolds in Euclidean Space
The Embedding Theorem
Embedding a Manifold with Boundary
Tangent Spaces and Critical Points
Tangent Lines
Critical Points
Nondegenerate Critical Points
A Stronger Embedding Theorem
Critical and Noncritical Levels
Definitions and Examples
A Neighborhood of a Critical Level; an Example
Neighborhood of a Critical Level; General Discussion
Neighborhood of a Critical Point
Neighborhood of a Critical Level; Summing Up
Spherical Modifications
Direct Embedding
Definition of Modifications
The Trace of a Modification
Cobounding Manifolds
Displacement and Isotopy
General Position
Rearrangement of Modifications
An Application to 3-Manifolds
Interpretation of Theorem 6-5 in Terms of Critical Points
Two-Dimensional Manifolds
Orientable 2-Manifolds
The Nonorientable Case
Second Steps
Killing of Homotopy Classes
Complementary Modifications and Cancellation
A Theorem on 3-Manifolds
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