On the chain-level intersection pairing for PL pseudomanifolds

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.

Keywords

intersection homology, intersection product, general position, pseudomanifold, differential graded algebra

2010 Mathematics Subject Classification

Primary 55N33, 55N45, 57Q65. Secondary 57N80.

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Published 1 January 2009