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

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

2010-10-03

arxiv.org/abs/1010.0402v1

Mathematics

Algebraic Topology

21 pages

Scientific paper

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 $\partial M$.

Affiliated with

Also associated with

No associations

Rating

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-512352