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

Al-Zamil, Qusay and Montaldi, James (2010) Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. [MIMS Preprint]

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Abstract

In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to Neumann (DN) operator $\Lambda$ on all forms on the boundary and they prove that the real additive de Rham cohomology structure of the manifold in question is completely determined by $\Lambda$. This shows that the DN map $\Lambda$ inscribes into the list of objects of algebraic topology. 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 $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+\iota_{X_M}$ on invariant forms on $M$. The main purpose is to adapt Belishev and Sharafutdinov's boundary data to invariant forms in terms of the operator $d_{X_M}$ and its adjoint $\delta_{X_M}$. In other words, we define an operator $\Lambda_{X_M}$ on invariant forms on the boundary which we call the $X_M$-DN map and using this we recover the long exact $X_M$-cohomology sequence of the topological pair $(M,\partial M)$ from an isomorphism with the long exact sequence formed from our boundary data. We then show that $\Lambda_{X_M}$ completely determines the free part of the relative and absolute equivariant cohomology groups of $M$ when the set of zeros of the corresponding vector field $X_M$ is equal to the fixed point set $F$ for the $G$-action. In addition, we partially determine the mixed cup product (the ring structure) of $X_M$-cohomology groups from $\Lambda_{X_M}$. These results explain to what extent the equivariant topology of the manifold in question is determined by the $X_M$-DN map $\Lambda_{X_M}$. Finally, we illustrate the connection between Belishev and Sharafutdinov's boundary data on $\partial F$ and ours on $\partial M$.

Item Type:	MIMS Preprint
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:	03 Oct 2010
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1528

Available Versions of this Item

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

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