GKM-sheaves and nonorientable surface group representations

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 $\mathcal{F}_X$ over a GKM-hypergraph $\Gamma_X$. Under the condition that $X$ is equivariantly formal, the ring of global sections of $\mathcal{F}_X$ are identified with the equivariant cohomology, $H^*_T (X; \mathbb{C}) \cong H^0(\mathcal{F}_X)$. 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 $\mathcal{R}K := \mathrm{Hom}(\pi_1 (\Sigma),K)$ under conjugation by $K$, where $\Sigma$ is a nonorientable surface and $K$ is a compact connected Lie group. We prove that $\mathcal{R}_{SU(3)}$ is equivariantly formal for all $\Sigma$ and compute its equivariant cohomology ring. We also produce conjectural betti number formulas for some other Lie groups.