Abstract

We study the interactions between the Witten deformation of the de Rham exterior differentiation and topological invariants in two scenarios of proper Lie group actions. In the first scenario, we work on a closed oriented manifold admitting an action by a compact connected Lie group. Using a special Morse-Bott function invariant under the group action, we deform the de Rham exterior derivative and get the associated Witten Laplacian. Applying asymptotic analysis, we localize the kernel of the Witten Laplacian around the critical components of the invariant Morse-Bott function. Finally, we build the chain isomorphism between the invariant Thom-Smale complex and the invariant Witten instanton complex. In the second scenario, we work on an oriented noncompact manifold admitting a proper cocompact action by a Lie group. First, through the generalized mod 2 index map between real KK-groups, we deform the Dirac type operator associated with the de Rham exterior derivative and find the appropriate Witten deformation of the de Rham exterior derivative. Then, we show that the invariant cohomology associated with this Witten deformation can be used to compute the semi-characteristic of the manifold. Finally, we prove that the semi-characteristic vanishes when there are two independent invariant vector fields on the manifold.

Committee Chair

Xiang Tang

Committee Members

Alexander Seidel; Aliakbar Daemi; Jr-Shin Li; Renato Feres; Yanli Song

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

5-30-2025

Language

English (en)

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Mathematics Commons

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