Algebraic Topology

1. Front Matter

   Pages i-xviii

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2. Calculus in the Plane

   1. Front Matter

      Pages 1-1

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   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

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   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

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   2. De Rham Cohomology and the Jordan Curve Theorem

      • William Fulton

      Pages 63-77

   3. Homology

      • William Fulton

      Pages 78-93

5. Vector Fields

   1. Front Matter

      Pages 95-95

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   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

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   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

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   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

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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

• 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

• Mathematics Department, University of Chicago, Chicago, USA

  William Fulton