March 9, 1999
91
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Victor Guillemin, Catalin Zara
Differential Geometry
The one-skeleton of a G-manifold M is the set of points p in M where $\dimG_p \geq \dim G -1$; and M is a GKM manifold if the dimension of thisone-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such amanifold this one-skeleton has the structure of a ``labeled" graph, $(\Gamma,\alpha)$, 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 thatthis ring is a free module ...
March 21, 2020
88
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Ivan Contreras, Andrew Rosevear
Combinatorics
Algebraic Topology
Differential Geometry
Mathematical Physics
We introduce a graph-theoretical interpretation of an induced action ofAut$(\Gamma)$ in the discrete de Rham cohomology of a finite graph $\Gamma$.This action produces a splitting of Aut$(\Gamma)$ that depends on the cycles of$\Gamma$. We also prove some graph-theoretical analogues of standard results indifferential geometry, in particular, a graph version of Stokes' Theorem andthe Mayer-Vietoris sequence in cohomology.
December 18, 2001
Victor Guillemin, Tara Holm, Catalin Zara
Symplectic Geometry
Combinatorics
Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ isa 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 onedimensional. For such a manifold these sets of orbits have the structure of alabelled 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...
December 22, 2018
87
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Oliver Goertsches, Leopold Zoller
Differential Geometry
Algebraic Topology
This is a survey on the equivariant cohomology of Lie group actions onmanifolds, from the point of view of de Rham theory. Emphasis is put on thenotion of equivariant formality, as well as on applications to ordinarycohomology and to fixed points.
March 17, 2005
87
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S. Julianna Tymoczko
Algebraic Geometry
Algebraic Topology
This paper provides an introduction to equivariant cohomology and homologyusing the approach of Goresky, Kottwitz, and MacPherson. When a group G actssuitably on a variety X, the equivariant cohomology of X can be computed usingthe combinatorial data of a skeleton of G-orbits on X. We give both a geometricdefinition and the traditional definition of equivariant cohomology. We includea discussion of the moment map and an algorithm for finding a set of generatorsfor the e...
March 2, 2008
87
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R. John Klein, Bruce Williams
Algebraic Topology
This is the second in a series of papers. Here we develop here anintersection theory for manifolds equipped with an action of a finite group. Asin our previous paper, our approach will be homotopy theoretic, enabling us tocircumvent the specter of equivariant transversality. This theory has applications to embedding problems, equivariant fixed pointtheory and the problem of enumerating the periodic points of a self map of acompact smooth manifold.
June 22, 2008
86
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Victor Guillemin, Silvia Sabatini, Catalin Zara
Combinatorics
Differential Geometry
The equivariant cohomology ring of a GKM manifold is isomorphic to thecohomology ring of its GKM graph. In this paper we explore the implications ofthis fact for equivariant fiber bundles for which the total space and the basespace are both GKM and derive a graph theoretical version of the Leray-Hirschtheorem. Then we apply this result to the equivariant cohomology theory of flagvarieties.
August 15, 2016
86
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Chen He
Algebraic Topology
Let a torus $T$ act smoothly on a compact smooth manifold $M$. If therational equivariant cohomology $H^*_T(M)$ is a free $H^*_T(pt)$-module, thenaccording to the Chang-Skjelbred Lemma, it can be determined by the$1$-skeleton consisting of the $T$-fixed points and $1$-dimensional $T$-orbitsof $M$. When $M$ is an even-dimensional, orientable manifold with 2-dimensional1-skeleton, Goresky, Kottwitz and MacPherson gave a graphic description of theequivariant cohomology. In...
March 5, 2007
86
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Bo Chen, Zhi Lü
Algebraic Topology
In this paper we consider a class of connected closed $G$-manifolds with anon-empty finite fixed point set, each $M$ of which is totally non-homologousto zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With thehelp of the equivariant index, we give an explicit description of theequivariant cohomology of such a $G$-manifold in terms of algebra, so that wecan obtain analytic descriptions of ring isomorphisms among equivariantcohomology rings of such $G$...
April 12, 2005
85
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Alexandra Mozgova
Geometric Topology
A standard fact about two incompressible surfaces in an irreducible3-manifold is that one can move one of them by isotopy so that theirintersection becomes $\pi_1$-injective. By extending it on the maps of some3-dimensional $\mathbb{Z}_n$-manifolds into 4-manifolds, we prove that anyhomotopy equivalence of 4-dimensional graph-manifolds with reducedgraph-structures is homotopic to a diffeomorphism preserving the structures. Keywords: graph-manifold, $\pi_1$-injective $\...
