Homotopy algebra structures on twisted tensor products and string topology operations

Miller, Micah

doi:10.2140/agt.2011.11.1163

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Homotopy algebra structures on twisted tensor products and string topology operations

Micah Miller

Algebraic & Geometric Topology 11 (2011) 1163–1203

DOI: 10.2140/agt.2011.11.1163

Abstract

Given a $C_{\infty}$ coalgebra $C_{*}$ , a strict dg Hopf algebra $H_{*}$ and a twisting cochain $τ : C_{*} \to H_{*}$ such that $Im (τ) \subset Prim (H_{*})$ , we describe a procedure for obtaining an $A_{\infty}$ coalgebra on $C_{*} \otimes H_{*}$ . This is an extension of Brown’s work on twisted tensor products. We apply this procedure to obtain an $A_{\infty}$ coalgebra model of the chains on the free loop space $L M$ based on the $C_{\infty}$ coalgebra structure of $H_{*} (M)$ induced by the diagonal map $M \to M \times M$ and the Hopf algebra model of the based loop space given by $T (H_{*} (M) [- 1])$ . When $C_{*}$ has cyclic $C_{\infty}$ coalgebra structure, we describe an $A_{\infty}$ algebra on $C_{*} \otimes H_{*}$ . This is used to give an explicit (nonminimal) $A_{\infty}$ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal $G$ –bundles.

Keywords

string topology, loop product, twisting cochain, homotopy algebra, $A_\infty$, $C_\infty$ algebra

Mathematical Subject Classification 2000

Primary: 55P35, 55R99, 57N65, 57R22, 57M99

Secondary: 55Q33, 55Q32

References

Publication

Received: 14 June 2010

Revised: 31 January 2011

Accepted: 4 February 2011

Published: 11 April 2011

Authors

Micah Miller
Department of Mathematics The Graduate Center at CUNY 365 Fifth Ave New York NY 10016 USA

Volume 11, issue 2 (2011)