On the chain-level intersection pairing for $PL$ pseudomanifolds

Abstract

James McClure recently showed that the domain for the intersection pairing of $PL$ chains on a $PL$ manifold $M$ is a subcomplex of $C∗(M) ⊗ C∗(M)$ that is quasi-isomorphic to $C∗(M) ⊗ C∗(M)$ 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.