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

Guillemin, V.; Zara, C.

Duke Math. J. 107, No. 2, 283-349 (2001).

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.

Reviewer: Tomasz Rybicki (Kraków)

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

Keywords:

GKM-manifold; labeled graph; equivariant cohomology

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