GKM sheaves and nonorientable surface group representations

Details GKM sheaves and nonorientable surface group representations GKM sheaves and nonorientable surface group representations

2010-08-09

arxiv.org/abs/1008.1517v2

Mathematics

Algebraic Topology

49 pages, fixed some typos and added a reference

Scientific paper

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 show that R_{SU(5)} fails to be equivariantly formal in general, disproving a conjecture from the author's thesis.

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