AlZamil, Qusay and Montaldi, James (2012) Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. Topology and Applications, 159. pp. 823832. ISSN 01668641
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Abstract
In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Î� on a compact Riemannian manifold M with boundary â��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 + Î¹XM on invariant forms on M. The main purpose is to adapt Belishevâ��Sharafutdinovâ��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 Î�XM on invariant forms on the boundary which we call the XMDN map and using this we recover the XMcohomology groups from the generalized boundary data (â��M,Î�XM ). This shows that for a Zariskiopen subset of the Lie algebra, Î�XM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XMcohomology groups from Î�XM . These results explain to what extent the equivariant topology of the manifold in question is determined by Î�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:  http://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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