On the chain-level intersection pairing for PL manifolds

McClure, J E¹

¹ Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN 47907-2067, USA

Geometry & topology, Tome 10 (2006) no. 3, pp. 1391-1424.

Résumé

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.

Détail

DOI : 10.2140/gt.2006.10.1391

Keywords: intersection pairing, partial algebra, general position

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