#### Volume 10, issue 3 (2006)

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On the chain-level intersection pairing for PL manifolds

### J E McClure

Geometry & Topology 10 (2006) 1391–1424
 arXiv: math.QA/0410450
##### 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.

##### Keywords
intersection pairing, partial algebra, general position
Primary: 18D50
Secondary: 57Q65