Algebraic & Geometric Topology

Cap products in string topology

Hirotaka Tamanoi

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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.

Article information

Algebr. Geom. Topol., Volume 9, Number 2 (2009), 1201-1224.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P35: Loop spaces 55P35: Loop spaces

Batalin–Vilkovisky algebra cap product intersection product loop bracket loop product string topology Batalin–Vilkovisky algebra cap product intersection product loop bracket loop product string topology


Tamanoi, Hirotaka. Cap products in string topology. Algebr. Geom. Topol. 9 (2009), no. 2, 1201--1224. doi:10.2140/agt.2009.9.1201.

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