1-skeleta, Betti numbers, and equivariant cohomology

Abstract

The 1-skeleton of a $G$-manifold $M$ is the set of points $p ∈ M$, where $\dim G_{p}≥ \dim G−1$, and $M$ is a GKM manifold if the dimension of this 1-skeleton is 2. M. Goresky, R. Kottwitz, and R. MacPherson show that for such a manifold this 1-skeleton has the structure of a “labeled” graph, $(Γ,α)$, and that the equivariant cohomology ring of $M$ is isomorphic to the “cohomology ring” of this graph. Hence, if $M$ is symplectic, one can show that this ring is a free module over the symmetric algebra $\mathbb{S}(\mathfrak{g}^*)$, with $b_{2i}(Γ)$ generators in dimension $2 i, b_{2i}(Γ)$ being the “combinatorial” $2i$th Betti number of $Γ$. In this article we show that this “topological” result is, in fact, a combinatorial result about graphs.