Formality Theorem for Hochschild Cochains via Transfer |
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Authors: | Vasily Dolgushev |
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Institution: | (1) Department of Mathematics, University of California at Riverside, 900 Big Springs Drive, Riverside, CA 92521, USA;(2) Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208, USA |
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Abstract: | We construct a 2-colored operad Ger
∞ which, on the one hand, extends the operad Ger
∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy
algebras. We show that Tamarkin’s Ger
∞-structure on the Hochschild cochain complex C
•(A, A) of an A
∞-algebra A extends naturally to a Ger+¥{{\bf Ger}^+_{\infty}}-structure on the pair (C
•(A, A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer
of this Ger+¥{{\bf Ger}^+_{\infty}}-structure to the cohomology of the pair (C
•(A, A), A). We show that Ger+¥{{\bf Ger}^+_{\infty}} is a sub DG operad of the first sheet E
1(SC) of the homology spectral sequence for the Fulton–MacPherson version SC of Voronov’s Swiss Cheese operad. Finally, we
prove that the DG operads Ger+¥{{\bf Ger}^+_{\infty}} and E
1(SC) are non-formal. |
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Keywords: | |
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