Infinity structure of poincaré duality spaces

Abstract

We show that the complex C.X of rational simplicial chains on a compact and triangulated Poincar é duality space X of dimension d is an A∞ coalgebra with ∞ duality. This is the structure required for an A ∞ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology HH.+d (C.X,C.X) of the cochain algebra C.X with values in C.X has a BV structure. This implies, if X is moreover simply connected, that the shifted homology H.+dLX of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of ∞ structures. © 2007 Mathematical Sciences Publishers.