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2009 Cap products in string topology
Hirotaka Tamanoi
Algebr. Geom. Topol. 9(2): 1201-1224 (2009). DOI: 10.2140/agt.2009.9.1201

Abstract

Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth manifold M admits the structure of a Batalin–Vilkovisky (BV) algebra equipped with an associative product (loop product) and a Lie bracket (loop bracket). We show that the cap product is compatible with the above two products in the loop homology. Namely, the cap product with cohomology classes coming from M via the circle action acts as derivations on the loop product as well as on the loop bracket. We show that Poisson identities and Jacobi identities hold for the cap product action, turning H(M)(LM) into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.

Citation

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Hirotaka Tamanoi. "Cap products in string topology." Algebr. Geom. Topol. 9 (2) 1201 - 1224, 2009. https://doi.org/10.2140/agt.2009.9.1201

Information

Received: 24 June 2007; Revised: 30 April 2009; Accepted: 22 May 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1175.55009
MathSciNet: MR2519587
Digital Object Identifier: 10.2140/agt.2009.9.1201

Subjects:
Primary: 55P35, 55P35

Rights: Copyright © 2009 Mathematical Sciences Publishers

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