Duke Mathematical Journal

1-skeleta, Betti numbers, and equivariant cohomology

V. Guillemin and C. Zara

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The 1-skeleton of a $G$-manifold $M$ is the set of points $p ∈ M$, where $\dim G_{p}≥ \dim G−1$, 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, $(Γ,α)$, 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 a free module over the symmetric algebra $\mathbb{S}(\mathfrak{g}^*)$, with $b_{2i}(Γ)$ generators in dimension $2 i, b_{2i}(Γ)$ being the “combinatorial” $2i$th Betti number of $Γ$. In this article we show that this “topological” result is, in fact, a combinatorial result about graphs.

Article information

Duke Math. J., Volume 107, Number 2 (2001), 283-349.

First available in Project Euclid: 5 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D20: Momentum maps; symplectic reduction
Secondary: 55N91: Equivariant homology and cohomology [See also 19L47] 57S15: Compact Lie groups of differentiable transformations


Guillemin, V.; Zara, C. 1-skeleta, Betti numbers, and equivariant cohomology. Duke Math. J. 107 (2001), no. 2, 283--349. doi:10.1215/S0012-7094-01-10724-2. https://projecteuclid.org/euclid.dmj/1091736759

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