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Infinity structure of Poincaré duality spaces. With an appendix by Dennis Sullivan. (English) Zbl 1137.57025
This paper shows that the chain complex \(C_{\bullet}X\) of a compact and triangulated Poincaré duality space \(X\) is naturally an \(A_{\infty}\) coalgebra with an \(\infty\) duality. For such a space the shifted Hochschild cohomology \(HH^{\bullet + d}(C^{\bullet}X, C_{\bullet}X)\) is a BV algebra with unit, so that it has a differential \(\Delta\) whose deviation from being a derivation defines a a graded Lie algebra. Further if \(X\) is simply connected, then the shifted homology \(H_{\bullet + d} LX\) of the free loop space \(LX\) is naturally a BV algebra with unit. In an appendix Dennis Sullivan provides a scheme for the local construction of \(\infty\) structures.

MSC:
57P10 Poincaré duality spaces
57P05 Local properties of generalized manifolds
55U15 Chain complexes in algebraic topology
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