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

Qusay S A Al-Zamil, James Montaldi

    Research output: Contribution to journalArticlepeer-review

    49 Downloads (Pure)

    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. © 2011 Elsevier B.V.
    Original languageEnglish
    Pages (from-to)823-832
    Number of pages9
    JournalTopology and its Applications
    Volume159
    Issue number3
    DOIs
    Publication statusPublished - 15 Feb 2012

    Keywords

    • Algebraic topology
    • Cup product (ring structure)
    • Dirichlet to Neumann operator
    • Equivariant cohomology
    • Equivariant topology
    • Group actions

    Fingerprint

    Dive into the research topics of 'Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology'. Together they form a unique fingerprint.

    Cite this