Topology for Physicists

1. Front Matter

   Pages I-XI

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2. Background

   • Albert S. Schwarz

   Pages 1-17

3. Fundamental Concepts

   • Albert S. Schwarz

   Pages 19-32

4. The Degree of a Map

   • Albert S. Schwarz

   Pages 33-43

5. The Fundamental Group and Covering Spaces

   • Albert S. Schwarz

   Pages 45-57

6. Manifolds

   • Albert S. Schwarz

   Pages 59-75

7. Differential Forms and Homology in Euclidean Space

   • Albert S. Schwarz

   Pages 77-100

8. Homology and Cohomology

   • Albert S. Schwarz

   Pages 101-157

9. Homotopy Classification of Maps of the Sphere

   • Albert S. Schwarz

   Pages 159-166

10. Homotopy Groups

    • Albert S. Schwarz

    Pages 167-172

11. Fibered Spaces

    • Albert S. Schwarz

    Pages 173-183

12. Fibrations and Homotopy Groups

    • Albert S. Schwarz

    Pages 185-189

13. Homotopy Theory of Fibrations

    • Albert S. Schwarz

    Pages 191-207

14. Lie Groups

    • Albert S. Schwarz

    Pages 209-216

15. Lie Algebras

    • Albert S. Schwarz

    Pages 217-231

16. Topology of Lie Groups and Homogeneous Manifolds

    • Albert S. Schwarz

    Pages 233-242

17. Geometry of Gauge Fields

    • Albert S. Schwarz

    Pages 243-285

18. Back Matter

    Pages 287-299

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In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. Topology has profound relevance to quantum field theory-for example, topological nontrivial solutions of the classical equa­ tions of motion (solitons and instantons) allow the physicist to leave the frame­ work of perturbation theory. The significance of topology has increased even further with the development of string theory, which uses very sharp topologi­ cal methods-both in the study of strings, and in the pursuit of the transition to four-dimensional field theories by means of spontaneous compactification. Im­ portant applications of topology also occur in other areas of physics: the study of defects in condensed media, of singularities in the excitation spectrum of crystals, of the quantum Hall effect, and so on. Nowadays, a working knowledge of the basic concepts of topology is essential to quantum field theorists; there is no doubt that tomorrow this will also be true for specialists in many other areas of theoretical physics. The amount of topological information used in the physics literature is very large. Most common is homotopy theory. But other subjects also play an important role: homology theory, fibration theory (and characteristic classes in particular), and also branches of mathematics that are not directly a part of topology, but which use topological methods in an essential way: for example, the theory of indices of elliptic operators and the theory of complex manifolds.

• Cohomology
• Fundamental group
• Gauge theory
• Homotopy
• Homotopy group
• condensed matter
• fibrations
• homology
• homotopy theory
• instantons
• manifold
• quantum field theory
• topology

• Department of Mathematics 565 Kerr Hall, University of California, Davis, USA

  Albert S. Schwarz

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