GKM sheaves and nonorientable surface group representations

TL;DR

This paper introduces GKM-sheaves as a unifying framework for GKM-theory and applies it to analyze the equivariant topology of representation varieties of nonorientable surface groups, including explicit cohomology computations.

Contribution

It develops the concept of GKM-sheaves over hypergraphs and applies this to study equivariant cohomology of nonorientable surface group representations, providing new computational tools.

Findings

R_{SU(3)} is equivariantly formal for all nonorientable surfaces

Computed the equivariant cohomology of R_{SU(3)}

Proposed conjectural Betti number formulas for other Lie groups

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(\pi_1(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.