GKM sheaves and nonorientable surface group representations

2010/08/09 by Baird, Thomas · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1008.1517

Abstract

Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global sections of F_X are identified with the equivariant cohomology, H_T^*(X; C). We show that GKM-sheaves provide a general framework able to incorporate numerous constructions in the GKM-theory literature. In the second half of the paper we apply these ideas to study the equivariant topology of the representation variety R_K := Hom(π₁(S), K) under conjugation by K, where S is a nonorientable surface and K is a compact connected Lie group. We prove that R_SU(3) is equivariantly formal for all S and compute its equivariant cohomology. . We also produce conjectural betti number formulas for some other Lie groups.

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