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2010.86: Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

2010.86: Qusay Al-Zamil and James Montaldi (2012) Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. Topology and Applications, 159. pp. 823-832. ISSN 0166-8641

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DOI: doi:10.1016/j.topol.2011.11.052

Abstract

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM = d + ιXM on invariant forms on M. The main purpose is to adapt Belishev–Sharafutdinov’s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM ). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM . These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM .

Item Type:Article
Uncontrolled Keywords:Algebraic Topology, equivariant topology, manifolds with boundary, equivariant cohomology, cup product (ring structure), group actions, Dirichlet to Neumann operator.
Subjects:MSC 2000 > 35 Partial differential equations
MSC 2000 > 55 Algebraic topology
MSC 2000 > 58 Global analysis, analysis on manifolds
MIMS number:2010.86
Deposited By:Dr James Montaldi
Deposited On:24 December 2011

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