Poisson structures on the homology of the space of knots

Details Poisson structures on the homology of the space of knots Poisson structures on the homology of the space of knots

2006-08-14

arxiv.org/abs/math/0608326v5

Geom. Topol. Monogr. 13 (2008) 463-482

Mathematics

Algebraic Topology

This is the version published by Geometry & Topology Monographs on 19 March 2008

Scientific paper

10.2140/gtm.2008.13.463

In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action of little disks operad, which was found by McClure and Smith. We show that the Browder operation coincides with the Gerstenhaber bracket on the Hochschild homology, which appears as the E2-term of the homology spectral sequence constructed by Bousfield. In particular we consider a variant of the space of long knots in higher dimensional Euclidean space, and show that Sinha's homology spectral sequence computes the Poisson algebra structure of the homology of the space. The Browder operation produces a homology class which does not directly correspond to chord diagrams.

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