Infinity Structure of Poincare Duality Spaces

TL;DR

This paper demonstrates that the chain complex of a Poincaré duality space has an A-infinity coalgebra structure with duality, leading to BV structures on Hochschild cohomology and free loop space homology.

Contribution

It establishes an A-infinity coalgebra with duality for Poincaré duality spaces, enabling BV structures on Hochschild cohomology and free loop space homology.

Findings

Chain complex of Poincaré duality space has A-infinity coalgebra structure.

Hochschild cohomology acquires a BV structure.

Free loop space homology admits a BV structure if space is simply connected.

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.