equivariant cohomology

equivariant cohomology
Name	equivariant cohomology
Field	Algebraic topology

equivariant cohomology

Equivariant cohomology is a tool in algebraic topology and differential geometry that studies spaces equipped with group actions, connecting ideas from Leray, Armand Borel, and Atiyah–Bott style localization. Originating in the work associated with Hirzebruch, Borel, Atiyah, and Bott, equivariant cohomology provides invariants for actions of compact Lie groups such as Lie groups, including SO(3), SU(2), and U(1), and plays a role across modern mathematics tied to figures like Deligne, Grothendieck, and Witten.

Introduction

Equivariant cohomology arose historically from attempts by Leray and Henri Cartan to combine cohomology theories with symmetry from groups like tori and Weyl group actions; later developments by Borel and applications by Michael Atiyah and Raoul Bott connected it to index theory and fixed-point formulas used by Edward Witten in quantum field theory. The subject sits at the intersection of work by Serre, Hirzebruch, Milnor, René Thom, and Marston Morse techniques, and relates to geometric representation theory advanced by Kazhdan, Lusztig, and Beilinson.

Definitions and Models

Several equivalent models define equivariant cohomology, inspired by constructions of Borel and formalized using classifying spaces like EG and BG. One model uses the homotopy quotient X_G = X ×_G EG linking to classifying space constructions familiar from John Milnor and Stasheff, while another algebraic model employs Cartan's differential model connecting with Koszul complex ideas used by André Weil and Henri Cartan. For compact groups like SO(n), SU(n), and U(n), the Cartan model uses invariant polynomials on the Lie algebra as in work by Chevalley and Eilenberg–MacLane techniques, and equivariant de Rham cohomology relates to the de Rham theorem of Georges de Rham.

Basic Properties and Examples

Basic properties mirror ordinary cohomology but incorporate group symmetries studied by Noether, Weyl, and David Hilbert style invariant theory. For a trivial action the equivariant cohomology reduces to the cohomology of the product with a classifying space as in Borel’s original formulation; for free actions it relates to quotients considered by Henri Poincaré and René Thom. Classic examples include actions on spheres studied by Heinz Hopf and projective spaces studied by real projective space and Complex projective space in the tradition of Hirzebruch and Shiing-Shen Chern. Equivariant cohomology of flag varieties ties to work by Bruhat, Chevalley, and Bott while toric variety examples connect to Delzant and William Fulton.

Localization and Equivariant Integration

Localization theorems, including results of Atiyah and Bott and the Atiyah–Bott localization theorem, reduce global equivariant data to fixed-point contributions, following ideas resonant with the Lefschetz fixed-point theorem of Solomon Lefschetz and the Atiyah–Singer index theorem developed by Atiyah and Isadore Singer. The Berline–Vergne formula and the ABBV localization formula connect to equivariant integration in the spirit of integrals studied by Gauss and Leonhard Euler, and have been used in computations involving moment maps from work by Atiyah, Guillemin, and Shlomo Sternberg.

Applications in Geometry and Topology

Equivariant cohomology applies to symplectic geometry as developed by Vladimir Arnold, Mikhail Gromov, and Dusa McDuff, and to moduli problems explored by Deligne and David Mumford. It underpins computations in Schubert calculus associated with Hermann Schubert and connects to mirror symmetry themes advanced by Maxim Kontsevich and Strominger, Shing-Tung Yau. In gauge theory and instanton moduli spaces it interacts with techniques from Simon Donaldson and Edward Witten, and in algebraic geometry it is used in intersection theory developed by Fulton and Grothendieck.

Computational Techniques and Calculations

Computational methods use spectral sequences related to works by Jean-Pierre Serre and Samuel Eilenberg, equivariant cell decompositions akin to those used by J. H. C. Whitehead and algorithms inspired by Schubert calculus and Alfred Young combinatorics. For torus actions, moment polytope techniques from Delzant and polyhedral methods in the style of Richard Stanley are effective; computations on flag varieties use Demazure operators developed by Michel Demazure and connections to Hecke algebra research by Kazhdan and Lusztig. Software implementations draw on symbolic systems influenced by the tradition of Francis Macaulay and computational algebra work by Buchberger.

Connections to Representation Theory and Physics

Equivariant cohomology links to representation theory through the study of fixed points and weights associated with actions of groups like GL(n), SL(n), and Sp(n), echoing themes from Hermann Weyl and Harish-Chandra. The interplay with geometric representation theory is seen in the work of Beilinson, Joseph Bernstein, and Victor Ginzburg, and it impacts quantum field theory and string theory via localization techniques used by Edward Witten, Nati Seiberg, and Cumrun Vafa. Equivariant K-theory and elliptic cohomology connections involve research by Atiyah, Graham Segal, and Landweber and appear in dualities explored by Kontsevich and Strominger–Yau–Zaslow.

Category:Algebraic topology