Simplicial intersection homology revisited

TL;DR

This paper establishes a direct quasi-isomorphism between simplicial and singular intersection homology complexes, extending the equivalence to blown-up cohomology and providing computational tools for pseudomanifolds and PL spaces.

Contribution

It proves a direct quasi-isomorphism between simplicial and singular intersection chains without using PL as an intermediate, and extends this to blown-up cohomology for pseudomanifolds and PL spaces.

Findings

The canonical map between simplicial and singular intersection chains is a quasi-isomorphism.

Blown-up intersection cohomology is isomorphic to singular cohomology for pseudomanifolds.

A new blown-up intersection cohomology for PL spaces is introduced and shown to be isomorphic to the singular version.

Abstract

Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the simplicial situations of intersection homology, uses the PL case as an intermediate. Here we show directly that the canonical map between the simplicial and the singular intersection chains complexes is a quasi-isomorphism. This is similar to the classical proof for simplicial complexes, with an argument based on the concept of residual complex and not on skeletons. This parallel between simplicial and singular approaches is also extended to the intersection blown-up cohomology that we introduced in a previous work. In the case of an orientable pseudomanifold, this cohomology owns a Poincar\'e isomorphism with the intersection homology, for any coefficient ring, thanks to a cap product with a fundamental class. So, the blown-up intersection cohomology of a pseudomanifold can be computed from a triangulation. Finally, we introduce a blown-up intersection cohomology for PL spaces and prove that it is isomorphic to the singular one.