Infinity structure of Poincaré duality spaces

Abstract

We show that the complex $C_{\bullet }X$ of rational simplicial chains on a compact and triangulated Poincaré duality space $X$ of dimension $d$ is an $A_{\infty }$ coalgebra with $\infty$ duality. This is the structure required for an A${}_{\infty }$ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology $H{H}^{\bullet +d}\left(C^{\bullet }X,C_{\bullet }X\right)$ of the cochain algebra $C^{\bullet }X$ with values in $C_{\bullet }X$ has a BV structure. This implies, if $X$ is moreover simply connected, that the shifted homology $H_{\bullet +d}LX$ of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of $\infty$ structures.