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 p¯(1)=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∗p¯(X)≅H∗p¯(X∗) for any perversity p¯ with the same growth conditions verifying p¯(1)≥0. In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, p¯, which corresponds to the classical topological invariance if 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.