Translator Disclaimer
1 April 2001 1-skeleta, Betti numbers, and equivariant cohomology
V. Guillemin, C. Zara
Duke Math. J. 107(2): 283-349 (1 April 2001). DOI: 10.1215/S0012-7094-01-10724-2

Abstract

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.

Citation

Download Citation

V. Guillemin. C. Zara. "1-skeleta, Betti numbers, and equivariant cohomology." Duke Math. J. 107 (2) 283 - 349, 1 April 2001. https://doi.org/10.1215/S0012-7094-01-10724-2

Information

Published: 1 April 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1020.57013
MathSciNet: MR1823050
Digital Object Identifier: 10.1215/S0012-7094-01-10724-2

Subjects:
Primary: 53D20
Secondary: 55N91 , 57S15

Rights: Copyright © 2001 Duke University Press

JOURNAL ARTICLE
67 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.107 • No. 2 • 1 April 2001
Back to Top