## 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

### Keywords:

GKM-manifold; labeled graph; equivariant cohomology
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### References:

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