## Algebraic & Geometric Topology

### Infinity structure of Poincaré duality spaces

#### Abstract

We show that the complex $C∙X$ of rational simplicial chains on a compact and triangulated Poincaré 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 1 (2007), 233-260.

Dates
Revised: 16 September 2006
Accepted: 22 January 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796666

Digital Object Identifier
doi:10.2140/agt.2007.7.233

Mathematical Reviews number (MathSciNet)
MR2308943

Zentralblatt MATH identifier
1137.57025

Subjects
Primary: 57P10: Poincaré duality spaces
Secondary: 57P05: Local properties of generalized manifolds

#### Citation

Tradler, Thomas; Zeinalian, Mahmoud. Infinity structure of Poincaré duality spaces. Algebr. Geom. Topol. 7 (2007), no. 1, 233--260. doi:10.2140/agt.2007.7.233. https://projecteuclid.org/euclid.agt/1513796666

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