On the chain-level intersection pairing for PL pseudomanifolds

TL;DR

This paper extends McClure's intersection pairing results from PL manifolds to PL stratified pseudomanifolds, establishing a partial commutative DGA structure and connecting it with sheaf-theoretic intersection homology.

Contribution

It generalizes the intersection pairing to PL stratified pseudomanifolds and constructs a sheaf-theoretic representation consistent with geometric intersection pairing.

Findings

Established a partial restricted commutative DGA structure for intersection chains.

Connected the sheaf-theoretic intersection pairing with the geometric Goresky-MacPherson pairing.

Extended McClure's results to a broader class of spaces.

Abstract

James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold $M$ is a subcomplex of $C_*(M)\otimes C_*(M)$ that is quasi-isomorphic to $C_*(M)\otimes C_*(M)$ and, more generally, that the intersection pairing endows $C_*(M)$ with the structure of a partially-defined commutative DGA. We generalize this theorem to intersection pairings of PL intersection chains on PL stratified pseudomanifolds and demonstrate the existence of a partial restricted commutative DGA structure. This structure is shown to generalize the iteration of the Goresky-MacPherson intersection product. As an application, we construct an explicit "roof" representation of the intersection homology pairing in the derived category of sheaves and verify that this sheaf theoretic pairing agrees with that arising from the geometric Goresky-MacPherson intersection pairing.