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1-skeleta, Betti numbers, and equivariant cohomology. (English) Zbl 1020.57013

Let \(G\) be a commutative compact connected Lie group acting on a manifold \(M\). \(M\) is called a GKM manifold if \(M^G\) is finite, \(M\) admits a \(G\)-invariant almost complex structure and its 1-skeleton (the set of all \(p\in M\) with \(\dim G_p\geq \dim G-1\)) is of \(\dim 2\). In view of a theorem due to Goresky, Kottwitz and MacPherson, the 1-skeleton has the structure of a labeled graph \((\Gamma,\alpha)\), and the equivariant cohomology \(H_G(H)\) can be reconstructed from the cohomology ring \(H(\Gamma,\alpha)\) of this graph.
The problem investigated in the paper is wether “topological” results about \(H(\Gamma,\alpha)\) are of a purely combinatorial nature. Two types of GKM manifolds with an affirmative answer to this problem constitute toric varieties and flag varieties. For further studies the authors define an abstract 1-skeleton to be a labeled graph \((\Gamma,\alpha)\) for which \(\alpha\) satisfies simple axioms. Next, the cohomology ring of it is introduced and computed, and a link with the concept of symplectic reduction is pointed out. Some applications are given, e.g. concerning Schubert polynomials. A realization theorem for abstract GKM-skeleta is shown.

MSC:

57S25 Groups acting on specific manifolds
05C90 Applications of graph theory
55N91 Equivariant homology and cohomology in algebraic topology
53D20 Momentum maps; symplectic reduction
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