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

Al-Zamil, Qusay and Montaldi, James (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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Abstract

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator � 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 �XM 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,�XM ). This shows that for a Zariski-open subset of the Lie algebra, �XM 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 �XM . These results explain to what extent the equivariant topology of the manifold in question is determined by �XM .

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 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology MSC 2010, the AMS's Mathematics Subject Classification > 58 Global analysis, analysis on manifolds
Depositing User:	Dr James Montaldi
Date Deposited:	24 Dec 2011
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1741

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• Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. (deposited 03 Oct 2010)
  • Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. (deposited 24 Dec 2011) [Currently Displayed]

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