On the chain-level intersection pairing for PL manifolds

Abstract

Let M be a compact oriented PL manifold and let C∗M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex of C∗M⊗C∗M. We prove that the inclusion map from this subcomplex to C∗M⊗C∗M is a quasi-isomorphism. An analogous result is true for the domain of the iterated intersection pairing. Using this, we show that the intersection pairing gives C∗M a structure of partially defined commutative DGA, which in particular implies that C∗M is canonically quasi-isomorphic to an E∞ chain algebra.