Introductory lectures on equivariant cohomology / Loring W. Tu ; with appendices by Loring W. Tu and Alberto Arabia.

Format:

Book

Author/Creator:

Tu, Loring W., author.

Contributor:

Arabia, Alberto, contributor.

Series:

Annals of mathematics studies ; number 204.

Annals of mathematics studies ; number 204

Language:

English

Subjects (All):

Homology theory.

Algebraic topology.

Physical Description:

xx, 315 pages : illustrations ; 24 cm.

Place of Publication:

Princeton, New Jersey : Princeton University Press, 2020.

Summary:

"This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. 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, Loring Tu 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"-- Provided by publisher.

Contents:

Homotopy groups and CW complexes

Principal bundles

Homotopy quotients and equivariant cohomology

Universal bundles and classifying spaces

Spectral sequences

Equivariant cohomology of S² under rotation

A universal bundle for a compact lie group

General properties of equivariant cohomology

The lie derivative and interior multiplication

Fundamental vector fields

Basic forms

Integration on a compact connected lie group

Vector-valued forms

The Maurer-Cartan form

Connections on a principal bundle

Curvature on a principal bundle

Differential graded algebras

The Weil algebra and the Weil model

Circle actions

The Cartan model in general

Outline of a proof of the equivariant de Rham theorem

Localization in algebra

Free and locally free actions

The topology of a group action

Borel localization for a circle action

A crash course in representation theory

Integration of equivariant forms

Rationale for a localization formula

Localization formulas

Proof of the localization formula for a circle action

Some applications.

Notes:

Includes bibliographical references and index.

Other Format:

Online version: Tu, Loring W. Introductory lectures on equivariant cohomology

ISBN:

9780691191744

0691191743

9780691191751

0691191751

OCLC:

1145901963