Simplicial intersection homology revisited
David Chataur, Martin Saralegi-Aranguren, Daniel Tanré
TL;DR
The paper directly proves that, for a full filtered simplicial complex $K$, the simplicial and singular intersection homologies ${H^{\overline p}_*(K)}$ and ${H^{\overline p}_*(|K|)}$ are canonically isomorphic, circumventing the need for CS-set intermediates by employing a residual complex decomposition with clots. It develops the blown-up intersection cohomology, establishes cup and cap product structures, and proves a Poincaré-type duality via the cap product with a fundamental class for orientable pseudomanifolds. It then extends these equivalences to blown-up cohomology, proving isomorphisms between simplicial, singular, and PL versions without CS-set hypotheses. The results enable computation of blown-up cohomology from triangulations and unify the simplicial, singular, and PL viewpoints across both IH and blown-up IH theories, with broad implications for duality and triangulational computation.
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 Poincaré 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.