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Mathematics  2014 

A chain level Batalin-Vilkovisky structure in string topology via de Rham chains

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Abstract:

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on homology of the free loop space of a closed $C^\infty$-manifold. Namely, we propose a new chain model of the free loop space, and define an action of a certain chain model of the framed little disks operad on it, recovering the original BV structure on homology level. We also compare this structure to a solution of Deligne's conjecture for Hochschild cochain complexes of differential graded algebras. To define the chain model of the loop space, we introduce a notion of de Rham chains, which is a hybrid of singular chains and differential forms.

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