On the chain-level intersection pairing for PL manifolds

Abstract

Let $M$ be a compact oriented PL manifold and let $C_{\ast }M$ be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex of $C_{\ast }M\otimes C_{\ast }M$. We prove that the inclusion map from this subcomplex to $C_{\ast }M\otimes C_{\ast }M$ 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 $C_{\ast }M$ a structure of partially defined commutative DGA, which in particular implies that $C_{\ast }M$ is canonically quasi-isomorphic to an $E_{\infty }$ chain algebra.