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.
