1-skeleta, Betti Numbers, and Equivariant Cohomology

The 1-skeleton of a G-manifold M is the set of points p ∈ M , where dimGp ≥ dimG − 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, ( , α), and that the equivariant cohomology ring ofM 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 S(g∗), with b2i ( ) generators in dimension 2i, b2i ( ) being the “combinatorial” 2ith Betti number of . In this article we show that this “topological” result is, in fact, a combinatorial result about graphs. 0. Introduction Let G be a commutative, compact, connected, n-dimensional Lie group, let g be its Lie algebra, let M be a compact 2d-dimensional manifold, and let τ : G×M → M be a faithful action of G on M . We say that M is a GKM manifold if it has the following properties. (1) M is finite. (2) M possesses a G-invariant almost complex structure. (3) For every p ∈ M, the weights αi,p ∈ g∗, i = 1, . . . , d, (0.1) of the isotropy representation of G on TpM are pairwise linearly independent. There is an alternate way of formulating this third condition. Let M be a Gmanifold that satisfies the first two conditions, and define the 1-skeleton of M to be the set of points, p ∈ M , with dimGp ≥ n− 1. Then M satisfies the third condition if and only if its 1-skeleton consists of G-invariant submanifolds that are fixed-pointfree and G-invariant embedded 2-spheres, each of which contains exactly two fixed DUKE MATHEMATICAL JOURNAL Vol. 107, No. 2, c © 2001 Received 17 August 1999. Revision received 27 March 2000. 2000 Mathematics Subject Classification. Primary 57S25; Secondary 05C90, 55N91. Guillemin’s work supported by National Science Foundation grant number DMS-890771.