*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0022
- Subject:
- Mathematics, Educational Mathematics

This chapter offers an outline of a proof of the equivariant de Rham theorem. In 1950, Henri Cartan proved that the cohomology of the base of a principal G-bundle for a connected Lie group G can be ...
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This chapter offers an outline of a proof of the equivariant de Rham theorem. In 1950, Henri Cartan proved that the cohomology of the base of a principal G-bundle for a connected Lie group G can be computed from the Weil model of the total space. From Cartan's theorem it is not too difficult to deduce the equivariant de Rham theorem for a free action. Guillemin and Sternberg presents an algebraic proof of the equivariant de Rham theorem, although some details appear to be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of a different approach using the Mayer–Vietoris argument. A limitation of the Mayer–Vietoris argument is that it applies only to manifolds with a finite good cover. The chapter provides a proof of the general case with no restrictions on the manifold and with all the details.Less

This chapter offers an outline of a proof of the equivariant de Rham theorem. In 1950, Henri Cartan proved that the cohomology of the base of a principal *G*-bundle for a connected Lie group *G* can be computed from the Weil model of the total space. From Cartan's theorem it is not too difficult to deduce the equivariant de Rham theorem for a free action. Guillemin and Sternberg presents an algebraic proof of the equivariant de Rham theorem, although some details appear to be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of a different approach using the Mayer–Vietoris argument. A limitation of the Mayer–Vietoris argument is that it applies only to manifolds with a finite good cover. The chapter provides a proof of the general case with no restrictions on the manifold and with all the details.

*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.001.0001
- Subject:
- Mathematics, Educational Mathematics

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into ...
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Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah–Bott and Berline–Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, the book begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.Less

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah–Bott and Berline–Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, the book begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.