Relation between intersection homology and homotopy groups

TL;DR

This paper introduces a new definition of intersection homotopy groups using simplicial sets, explores their properties, and establishes their relationship with intersection homology, including invariance and specific cases.

Contribution

It defines intersection homotopy groups via simplicial sets, proves foundational theorems, and relates them to intersection homology, extending classical homotopy concepts to stratified spaces.

Findings

Established Van Kampen theorem for intersection fundamental groups.

Proved a Hurewicz theorem linking intersection homotopy groups and homology.

Showed invariance of intersection homotopy groups under certain conditions.

Abstract

As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates to a couple $(X,\overline{p})$ of a filtered space and a perversity. We first establish some basic properties for the intersection fundamental groups, as a Van Kampen theorem. For general intersection homotopy groups on Siebenmann CS sets, we prove a Hurewicz theorem between them and the Goresky and MacPherson intersection homology. If the CS set and its intrinsic stratification have the same regular part, we establish the topological invariance of the $\overline{p}$-intersection homotopy groups. Several examples justify the hypotheses made in the statements. Finally, intersection homotopy groups also coincide with the homotopy groups of the topological space itself, for the top perversity on a connected, normal Thom-Mather space.