Volume 9, issue 2 (2009)

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Cap products in string topology

Hirotaka Tamanoi

Algebraic & Geometric Topology 9 (2009) 1201–1224

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.

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
Received: 24 June 2007
Revised: 30 April 2009
Accepted: 22 May 2009
Published: 13 June 2009
Hirotaka Tamanoi
Department of Mathematics
University of California Santa Cruz
Santa Cruz, CA 95064