The 1-skeleton of a G-manifold M is the set of points p epsilon M, where dim G(p) greater than or equal to 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, (Gamma, alpha), 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 af ree module over the symmetric algebra S(g*), with b(2i)(Gamma) generators in dimension 2i, b(2i)(Gamma) being the "combinatorial" 2i th Betti number of Gamma. In this article we show that this "topological" result is, in fact, a combinatorial result about graphs. [References: 39]
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