On the chain-level intersection pairing for PL manifolds

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 $C_{*} M \otimes C_{*} M$ . We prove that the inclusion map from this subcomplex to $C_{*} M \otimes C_{*} M$ 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 $E_{\infty}$ chain algebra.

Keywords

intersection pairing, partial algebra, general position

Mathematical Subject Classification 2000

Primary: 18D50

Secondary: 57Q65

References

Forward citations

Publication

Received: 22 October 2004

Accepted: 14 July 2005

Published: 4 October 2006

Proposed: Gunnar Carlsson

Seconded: Haynes Miller, Bill Dwyer

Correction: 12 March 2009

Authors

J E McClure
Department of Mathematics Purdue University 150 N University Street West Lafayette, IN 47907-2067 USA

Volume 10, issue 3 (2006)

J E McClure

Abstract