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Poisson structures on the homology of the space of knots

Keiichi Sakai

Geometry & Topology Monographs 13 (2008) 463–482

DOI: 10.2140/gtm.2008.13.463

arXiv: math.AT/0608326

Abstract

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.

Keywords

space of long knots, little disks operad, Browder operation, Gerstenhaber bracket, McClure-Smith machinery

Mathematical Subject Classification

Primary: 55P48

Secondary: 55P35

References
Publication

Received: 19 September 2006
Revised: 16 July 2007
Accepted: 23 July 2007
Published: 19 March 2008

Authors
Keiichi Sakai
Graduate School of Mathematical Science
University of Tokyo
3-8-1 Komaba
Meguro
Tokyo 153-8914
Japan