Introductory Lectures on Equivariant Cohomology

This book is the fourth and most advanced in the series on geometry by the author, following An Introduction to Manifolds, Differential Forms in Algebraic Topology (with Raoul Bott), and Differential geometry: Connections, Curvature and Characteristic Classes. Equivariant theory studies spaces equipped with group actions. Since symmetries are ubiquitous in many areas of mathematics, equivariant theory has many applications in various areas of mathematics and physics. This book does not attempt to be an exhaustive account of equivariant theory, choosing instead to give a readable and relatively complete account of the chosen material. Given the track record of the author’s other excellent works, it should be no surprise that this book is extremely well written and organized. 

The book focuses on equivariant Borel cohomology and does not address other equivariant cohomology theories. It begins from a topological point of view, introducing equivariant cohomology using the homotopy quotient and bundle theory. It then gives background on the differential geometry of principal bundles. This material is used to give a detailed development of how to define equivariant cohomology using differential forms, covering both the Cartan and Weil models of equivariant cohomology, leading up to a proof of the equivariant de Rham theorem. It then discusses the idea of localization, and the equivariant localization formula and its implications. Actions of the circle group are presented as a running example throughout. The book is organized into many short sections, grouped into five overall parts. Most of these sections deliver a digestible portion of mathematical content. I particularly appreciated the sections which gave motivational exposition of the ideas covered by other more technical sections. 

The book assumes basic background on algebraic topology and differential geometry, and should be accessible to second year graduate students. The book does cover background material, but the level is inconsistent. Material on the structure of the differential forms of principal bundles used in defining equivariant cohomology is carefully and thoroughly presented, but other background like the section on spectral sequences is more cursory, and probably requires either prior knowledge or suspension of disbelief on the part of the reader. Likewise, the exercises presented are minimal and of widely varying difficulty. Therefore, it would require some additional material to use this book as a textbook for a complete course. But it serves very well as an overview for those looking to start learning about this material, who simply want to read a lucid account of a beautiful area of mathematics and are willing to fill in some details from other sources, or just go with the flow and enjoy the story.

Laura Scull is an Associate Professor of Mathematics at Fort Lewis College.