Intersection homology: General perversities and topological invariance

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 $\overline{p}\left(1\right)=\overline{p}\left(2\right)=0$. King reproves this invariance by associating an intrinsic pseudomanifold $X^{\ast }$ to any pseudomanifold $X$. His proof consists of an isomorphism between the associated intersection homologies $H_{\ast }^{\overline{p}}\left(X\right)\cong H_{\ast }^{\overline{p}}\left(X^{\ast }\right)$ for any perversity $\overline{p}$ with the same growth conditions verifying $\overline{p}\left(1\right)\ge 0$.

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 $X^{\ast }$. In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification.