The Witten deformation for even dimensional conformally conic manifolds

Abstract

The goal of this article is to generalise the Witten deformation to even dimensional conformally conic manifolds X X and a class of functions f : X → R f: X \to \mathbb R called admissible Morse functions. We get Morse inequalities relating the L 2 \mathrm {L}^2 -Betti numbers of X X with the number of critical points of the function f f . Hereby the contribution of a singular point p p of X X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse data of f f at p p . The definition of an admissible Morse function is inspired by stratified Morse theory as developed by Goresky and MacPherson.