A GKM description of the equivariant cohomology ring of a homogeneous space

2001/12/18 by Victor Guillemin, Guillemin, Victor, Tara Holm +3 · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.CO #math.SG #msc:05C25 #msc:53D05 #msc:55N91

paper · pdf · doi:10.48550/arxiv.math/0112184

19 pages, 3 figures

arxiv created 2001/12/18 · arxiv updated 2009/11/30

Abstract

Let T be a torus of dimension n>1 and M a compact T-manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/T is zero dimensional and the set of one dimensional orbits in M/T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph. In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.

Cited by

• Computations in equivariant Gromov-Witten theory of GKM spaces
• Extensions of realizable Hamiltonian and complexity one GKM₄ graphs
• Notes on the cohomology of partial Hessenberg varieties

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