1-skeleta, Betti numbers, and equivariant cohomology
V. Guillemin and C. Zara
Source: Duke Math. J. Volume 107, Number 2 (2001), 283-349.
Abstract
The 1-skeleton of a G-manifold M is the set
of points p∈M, where dim
Gp≥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 (
*), with
b2i(Γ) generators in dimension
2i,b2i(Γ) being the "combinatorial" 2ith Betti
number of Γ. In this article we show that this
"topological" result is, in fact, a combinatorial result about
graphs.
Primary Subjects: 53D20
Secondary Subjects: 55N91, 57S15
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Zentralblatt MATH identifier:
1020.57013
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