Intersection homotopy, refinements and coarsenings

TL;DR

This paper extends the theory of intersection homotopy groups to coarsenings of CS sets with general perversities, establishing invariance results under certain conditions and broadening the topological understanding of stratified spaces.

Contribution

It introduces a framework for analyzing coarsenings of CS sets with general perversities and proves invariance theorems for their intersection homotopy groups.

Findings

Invariance of intersection homotopy groups under coarsening with growing perversities.

Extension of invariance results to Thom-Mather spaces where singular strata become regular.

Generalization of perversity definitions beyond codimension-based to poset-based.

Abstract

In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged. Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one being a union of strata of the other. We endow them with a general perversity and its pushforward, where the adjective ``general'' means that the perversities are defined on the poset of the strata and not only according to their codimension. If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts. An invariance is shown too in some cases where singular strata become regular in the coarsening, for Thom-Mather spaces.