## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 12, Number 4 (2014), 867-921.

### 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.

#### Article information

**Source**

J. Symplectic Geom., Volume 12, Number 4 (2014), 867-921.

**Dates**

First available in Project Euclid: 1 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1433196065

**Mathematical Reviews number (MathSciNet)**

MR3333031

**Zentralblatt MATH identifier**

1329.55004

#### Citation

Baird, Thomas. GKM-sheaves and nonorientable surface group representations. J. Symplectic Geom. 12 (2014), no. 4, 867--921. https://projecteuclid.org/euclid.jsg/1433196065