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Differential Topology

  • Textbook
  • © 1976

Overview

Authors:
  1. Morris W. Hirsch

    1. Department of Mathematics, University of California, Berkeley, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 33)

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Table of contents (10 chapters)

  1. Front Matter

    Pages N2-x
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  2. Introduction

    • Morris W. Hirsch
    Pages 1-6

  3. Manifolds and Maps

    • Morris W. Hirsch
    Pages 7-33

  4. Function Spaces

    • Morris W. Hirsch
    Pages 34-66

  5. Transversality

    • Morris W. Hirsch
    Pages 67-84

  6. Vector Bundles and Tubular Neighborhoods

    • Morris W. Hirsch
    Pages 85-119

  7. Degrees, Intersection Numbers, and the Euler Characteristic

    • Morris W. Hirsch
    Pages 120-141

  8. Morse Theory

    • Morris W. Hirsch
    Pages 142-168

  9. Cobordism

    • Morris W. Hirsch
    Pages 169-176

  10. Isotopy

    • Morris W. Hirsch
    Pages 177-187

  11. Surfaces

    • Morris W. Hirsch
    Pages 188-208

  12. Back Matter

    Pages 209-221
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Keywords

About this book

This book presents some of the basic topological ideas used in studying differentiable manifolds and maps. Mathematical prerequisites have been kept to a minimum; the standard course in analysis and general topology is adequate preparation. An appendix briefly summarizes some of the back­ ground material. In order to emphasize the geometrical and intuitive aspects of differen­ tial topology, I have avoided the use of algebraic topology, except in a few isolated places that can easily be skipped. For the same reason I make no use of differential forms or tensors. In my view, advanced algebraic techniques like homology theory are better understood after one has seen several examples of how the raw material of geometry and analysis is distilled down to numerical invariants, such as those developed in this book: the degree of a map, the Euler number of a vector bundle, the genus of a surface, the cobordism class of a manifold, and so forth. With these as motivating examples, the use of homology and homotopy theory in topology should seem quite natural. There are hundreds of exercises, ranging in difficulty from the routine to the unsolved. While these provide examples and further developments of the theory, they are only rarely relied on in the proofs of theorems.

Reviews

M.W. Hirsch

Differential Topology

"A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology. Newly introduced concepts are usually well motivated, and often the historical development of an idea is described. There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text. "—MATHEMATICAL REVIEWS

Authors and Affiliations

  • Department of Mathematics, University of California, Berkeley, USA

    Morris W. Hirsch

Bibliographic Information

  • Book Title: Differential Topology

  • Authors: Morris W. Hirsch

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4684-9449-5

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1976

  • Hardcover ISBN: 978-0-387-90148-0Published: 01 July 1976

  • Softcover ISBN: 978-1-4684-9451-8Published: 28 June 2012

  • eBook ISBN: 978-1-4684-9449-5Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: X, 222

  • Topics: Manifolds and Cell Complexes (incl. Diff.Topology)

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