Algebraic Topology

A First Course

  • William Fulton

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

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Calculus in the Plane

    1. Front Matter
      Pages 1-1
    2. William Fulton
      Pages 3-16
    3. William Fulton
      Pages 17-31
  3. Winding Numbers

    1. Front Matter
      Pages 33-33
    2. William Fulton
      Pages 35-47
    3. William Fulton
      Pages 48-58
  4. Cohomology and Homology, I

    1. Front Matter
      Pages 59-61
    2. William Fulton
      Pages 78-93
  5. Vector Fields

    1. Front Matter
      Pages 95-95
    2. William Fulton
      Pages 97-105
    3. William Fulton
      Pages 106-119
  6. Cohomology and Homology, II

    1. Front Matter
      Pages 121-122
    2. William Fulton
      Pages 123-136
    3. William Fulton
      Pages 137-150
  7. Covering Spaces and Fundamental Groups, I

    1. Front Matter
      Pages 151-151
    2. William Fulton
      Pages 153-164
    3. William Fulton
      Pages 165-175
  8. Covering Spaces and Fundamental Groups, II

    1. Front Matter
      Pages 177-178

About this book

Introduction

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

  • William Fulton
    • 1
  1. 1.Mathematics DepartmentUniversity of ChicagoChicagoUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-4180-5
  • Copyright Information Springer-Verlag New York 1995
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-94327-5
  • Online ISBN 978-1-4612-4180-5
  • Series Print ISSN 0072-5285
  • About this book