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Batalin-Vilkovisky algebra structures on Hochschild cohomology. (English) Zbl 1180.16007

Let \(\mathbb{F}\) be an arbitrary field, \(M\) be a compact oriented \(d\)-dimensional smooth manifold and \(LM\) be the free loop space on \(M\). The author proves that, if \(M\) is simply connected, then the Gerstenhaber algebra structure on the Hochschild cohomology of the algebra of singular cochains of \(M\) extends to a Batalin-Vilkovisky algebra structure isomorphic to that on \(H_{*+d}(LM)\), the shifted free loop space homology on \(M\).

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16E45 Differential graded algebras and applications (associative algebraic aspects)
55P35 Loop spaces
57P10 Poincaré duality spaces
55P62 Rational homotopy theory
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