Comparison between two complexes on a singular space

Ludwig U

Abstract

In this article we generalise the Witten deformation for stratified spaces and a class of Morse functions which we call radial Morse functions. In the first part of the article we perform the Witten deformation on the complex of L 2 -forms on a (general) stratified space. A local Spectral Gap Theorem for the model Witten Laplacian near critical points of the Morse function is proved for general stratified spaces. The global Spectral Gap Theorem for the Witten Laplacian can be generalised to spaces with isolated singularities as well as to Witt spaces. In the second part of the article we give, in the case of isolated singularities, a geometric interpretation of the complex of eigenforms of the Witten Laplacian to small eigenvalues. For this, we have to establish first an analogue of the Morse–Thom–Smale complex.