#### Volume 9, issue 2 (2009)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Cap products in string topology

### Hirotaka Tamanoi

Algebraic & Geometric Topology 9 (2009) 1201–1224
##### 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 ${H}^{\ast }\left(M\right)\oplus {ℍ}_{\ast }\left(LM\right)$ into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.

##### Keywords
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
##### Mathematical Subject Classification 2000
Primary: 55P35, 55P35
##### Publication
Received: 24 June 2007
Revised: 30 April 2009
Accepted: 22 May 2009
Published: 13 June 2009
##### Authors
 Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz Santa Cruz, CA 95064