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  • Textbook
  • © 1995

Algebraic Topology

A First Course

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 153)

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  • ISBN: 978-1-4612-4180-5
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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xviii
  2. Calculus in the Plane

    1. Front Matter

      Pages 1-1
    2. Path Integrals

      • William Fulton
      Pages 3-16
    3. Angles and Deformations

      • William Fulton
      Pages 17-31
  3. Winding Numbers

    1. Front Matter

      Pages 33-33
    2. The Winding Number

      • William Fulton
      Pages 35-47
    3. Applications of Winding Numbers

      • William Fulton
      Pages 48-58
  4. Cohomology and Homology, I

    1. Front Matter

      Pages 59-61
    2. Homology

      • William Fulton
      Pages 78-93
  5. Vector Fields

    1. Front Matter

      Pages 95-95
    2. Indices of Vector Fields

      • William Fulton
      Pages 97-105
    3. Vector Fields on Surfaces

      • William Fulton
      Pages 106-119
  6. Cohomology and Homology, II

    1. Front Matter

      Pages 121-122
    2. Holes and Integrals

      • William Fulton
      Pages 123-136
    3. Mayer—Vietoris

      • William Fulton
      Pages 137-150
  7. Covering Spaces and Fundamental Groups, I

    1. Front Matter

      Pages 151-151
    2. Covering Spaces

      • William Fulton
      Pages 153-164
    3. The Fundamental Group

      • William Fulton
      Pages 165-175
  8. Covering Spaces and Fundamental Groups, II

    1. Front Matter

      Pages 177-178

About this book

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re­ lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ­ ential topology, etc.), we concentrate our attention on concrete prob­ lems in low dimensions, introducing only as much algebraic machin­ ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol­ ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel­ opment of the subject. What would we like a student to know after a first course in to­ pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under­ standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind­ ing numbers and degrees of mappings, fixed-point theorems; appli­ cations such as the Jordan curve theorem, invariance of domain; in­ dices of vector fields and Euler characteristics; fundamental groups

Keywords

  • algebra
  • algebraic curve
  • algebraic topology
  • cohomology
  • cohomology group
  • De Rham cohomology
  • Dimension
  • Euler characteristic
  • fixed-point theorem
  • Fundamental group
  • homology
  • Homotopy
  • Homotopy group
  • topology
  • Winding number

Authors and Affiliations

  • Mathematics Department, University of Chicago, Chicago, USA

    William Fulton

Bibliographic Information

  • Book Title: Algebraic Topology

  • Book Subtitle: A First Course

  • Authors: William Fulton

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-4180-5

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media, Inc. 1995

  • Softcover ISBN: 978-0-387-94327-5

  • eBook ISBN: 978-1-4612-4180-5

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XVIII, 430

  • Number of Illustrations: 13 b/w illustrations

  • Topics: Topology

Buying options

eBook USD 34.99
Price excludes VAT (USA)
  • ISBN: 978-1-4612-4180-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 49.95
Price excludes VAT (USA)