Infinity structure of Poincaré duality spaces

Tradler, Thomas; Zeinalian, Mahmoud

doi:10.2140/agt.2007.7.233

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Infinity structure of Poincaré duality spaces

Thomas Tradler and Mahmoud Zeinalian

Appendix: Dennis Sullivan

Algebraic & Geometric Topology 7 (2007) 233–260

DOI: 10.2140/agt.2007.7.233

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 $A_{\infty}$ 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 $H H^{∙ + d} (C^{∙} X, C_{∙} X)$ 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 $H_{∙ + d} L X$ 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

Mathematical Subject Classification 2000

Primary: 57P10

Secondary: 57P05

References

Publication

Received: 21 August 2005

Revised: 16 September 2006

Accepted: 22 January 2007

Published: 29 March 2007

Authors

Thomas Tradler
Department of Mathematics College of Technology of the City University of New York 300 Jay Street Brooklyn NY 11201 USA

Mahmoud Zeinalian
Department of Mathematics C W Post Campus of Long Island University 720 Northern Boulevard Brookville NY 11548 USA

Dennis Sullivan
Department of Mathematics Graduate Center of the City University of New York 365 Fifth Avenue New York NY 10016 USA

Volume 7, issue 1 (2007)