Infinity Structure of Poincare Duality Spaces

Abstract

We show that the complex $C_\bullet X$ of rational simplicial chains on a compact and triangulated Poincar\'e 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 $HH^{\bullet+d} (C^\bullet X, C_\bullet X)$ 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.