Poincaré-Hopf theorems on singular spaces

Abstract

In 1991, Schwartz proved a Poincaré–Hopf theorem for stratified radial vector fields on Whitney stratified analytic varieties with boundary. In this paper, we extend her theory and results to a wide class of stratified sets with boundary, called radial manifold complexes, which include all Thom–Mather stratified sets, but also arbitrary closure orderable finite partitions of subanalytic sets into subanalytic submanifolds. In the process, we give a short proof of a generalization of the Poincaré–Hopf theorem for radial vector fields of Schwartz (our Theorem 3.4). We then study more general stratified vector fields that need not be radial and may further be tangent to the boundary of the manifold complex, and prove the corresponding Poincaré–Hopf theorems. In particular, a notion is given of the virtual index of a vector field on a radial manifold complex and the Euler characteristic of the complex is shown to be the sum of the virtual indices.