The mod 2 cohomology of fixed point sets of anti-symplectic involutions

2001/07/20 by Daniel Biss, Biss, Daniel, Victor W. Guillemin +3 · 1 citation
Mathematics · #53D05 (primary) #55N91 (secondary) #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AT #math.DG #math.SG #msc:53D05 #msc:55N91

paper · pdf · doi:10.48550/arxiv.math/0107151

21 pages, 1 figure

arxiv created 2003/09/10 · arxiv updated 2009/11/30

Abstract

Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus G=Tⁿ. Suppose that σ is an anti-symplectic involution compatible with the G-action. The real locus of M is X, the fixed point set of σ. Duistermaat uses Morse theory to give a description of the ordinary cohomology of X in terms of the cohomology of M. There is a residual \G=(\Zt)ⁿ action on X, and we can use Duistermaat's result, as well as some general facts about equivariant cohomology, to prove an equivariant analogue to Duistermaat's theorem. In some cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and Goldin-Holm to the real locus.

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