#### Volume 7, issue 1 (2007)

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Infinity structure of Poincaré duality spaces

### Appendix: Dennis Sullivan

Algebraic & Geometric Topology 7 (2007) 233–260
##### 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.

##### Keywords
Poincaré duality space, local infinity structure
Primary: 57P10
Secondary: 57P05