The chain-level intersection pairing for PL pseudomanifolds revisited

TL;DR

This paper extends the PL intersection product to non-compact spaces, enabling sheafification and duality proofs in intersection homology, with corrections to foundational duality results for pseudomanifolds.

Contribution

It generalizes the intersection product for chains on PL pseudomanifolds to non-compact spaces and corrects key duality proofs in the theory.

Findings

Extended intersection product to non-compact spaces

Sheafification of the intersection product achieved

Provided correction to Goresky-MacPherson Poincaré duality proof

Abstract

We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential step in proving duality between the Goresky-MacPherson intersection homology product and the intersection cohomology cup product pairing recently defined by the author and McClure. We also provide a correction to the Goresky-MacPherson proof of a version of Poincar\'e duality on pseudomanifolds that is used in the construction of the intersection product.