Cap Products in String Topology

Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth finite dimensional manifold M admits the structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative product called the loop product and a Lie bracket called the 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 loop products as well as on loop brackets. We show that Poisson identities and Jacobi identities hold for the cap product action, extending the BV structure in the loop homology to the one including the cohomology of M. Finally, we describe the cap product in terms of the BV algebra structure in the loop homology.

💡 Research Summary

The paper investigates how the cap product interacts with the algebraic structures on the homology of the free loop space (LM=\operatorname{Map}(S^{1},M)) introduced by Chas and Sullivan. For an oriented closed smooth manifold (M) of dimension (n), Chas–Sullivan showed that (H_{*}(LM)) carries a Batalin‑Vilkovisky (BV) algebra: a degree (-n) associative product (the loop product (\bullet)), a degree (-1) operator (\Delta) whose failure to be a derivation yields a Lie bracket ({,}) (the loop bracket), and the usual BV identities.

The authors focus on cohomology classes of (M) and their images in (H^{}(LM)) via the natural (S^{1})-action on loops. Given (a\in H^{}(M)), let (\tilde a\in H^{}(LM)) denote its transfer. For any chain (x\in H_{}(LM)) the cap product (\tilde a\cap x) is defined. The main results are:

1. Derivation property for the loop product
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