Cap products in string topology

Chas and Sullivan showed that the homology of the free loop space $L M$ 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 $H^{*} (M) \oplus ℍ_{*} (L M)$ into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.

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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

References

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

Volume 9, issue 2 (2009)

Hirotaka Tamanoi

Abstract