Intersection Homology. General perversities and topological invariance

TL;DR

This paper extends the topological invariance of intersection homology to strata-dependent perversities and tame intersection homology, broadening the class of invariances beyond classical perversities and stratification refinements.

Contribution

It proves topological invariance for strata-dependent perversities and extends invariance results to tame intersection homology under certain conditions.

Findings

Invariance holds for strata-dependent perversities matching classical cases.

Tame intersection homology is invariant for large perversities without singular strata.

Invariance persists under stratification refinement in specified conditions.

Abstract

Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent perversities with some growth conditions, verifying $\overline p(1)=\overline p(2)=0$. King reproves this invariance by associating an intrinsic pseudomanifold $X^*$ to any pseudomanifold $X$. His proof consists of an isomorphism between the associated intersection homologies $H^{\overline{p}}_{*}(X) \cong H^{\overline{p}}_{*}( X^*)$ for any perversity $\overline{p}$ with the same growth conditions verifying $\overline p(1)\geq 0$. In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, $\overline{p}$, which corresponds to the classical topological invariance if $\overline{p}$ is a GM-perversity. We also extend it to the tame intersection homology, a variation of the intersection homology, particularly suited for ``large'' perversities, if there is no singular strata on $X$ becoming regular in $X^*$. In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification.