## Algebraic & Geometric Topology

### Cap products in string topology

Hirotaka Tamanoi

#### 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 $H∗(M)⊕ℍ∗(LM)$ into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.

#### Article information

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

Dates
Revised: 30 April 2009
Accepted: 22 May 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513797012

Digital Object Identifier
doi:10.2140/agt.2009.9.1201

Mathematical Reviews number (MathSciNet)
MR2519587

Zentralblatt MATH identifier
1175.55009

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

#### Citation

Tamanoi, Hirotaka. Cap products in string topology. Algebr. Geom. Topol. 9 (2009), no. 2, 1201--1224. doi:10.2140/agt.2009.9.1201. https://projecteuclid.org/euclid.agt/1513797012

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