Witten’s perturbation and Lefschetz formula on singular spaces

Abstract

The objects studied in this thesis are Thom-Mather stratified spaces. Thom-Mather stratified spaces admit a partition into smooth manifolds called strata, which in general have different dimensions. The strata are glued under certain technical conditions involving conic bundles. This gives rise to a local description of these spaces using conical charts, which generalize the usual charts on smooth manifolds. Moreover several kinds of metrics can be defined on the strata of Thom-Mather stratified spaces: general adapted metrics, adapted metrics and adapted metrics of conic type.

Some differential operators can be considered on strata. Their study is a powerful technique to obtain properties of stratified spaces. Thus Functional Analysis and Partial Differential Equations, particularly the heat equation and the wave equation, are fundamental tools in this field. The mathematical area that applies Operator Theory in order to obtain geometrical and topological results on manifolds and related objects is called Global Analysis.