#### Volume 11, issue 2 (2011)

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Homotopy algebra structures on twisted tensor products and string topology operations

### Micah Miller

Algebraic & Geometric Topology 11 (2011) 1163–1203
##### Abstract

Given a ${C}_{\infty }$ coalgebra ${C}_{\ast }$, a strict dg Hopf algebra ${H}_{\ast }$ and a twisting cochain $\tau :{C}_{\ast }\to {H}_{\ast }$ such that $Im\left(\tau \right)\subset Prim\left({H}_{\ast }\right)$, we describe a procedure for obtaining an ${A}_{\infty }$ coalgebra on ${C}_{\ast }\otimes {H}_{\ast }$. 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 $LM$ based on the ${C}_{\infty }$ coalgebra structure of ${H}_{\ast }\left(M\right)$ induced by the diagonal map $M\to M×M$ and the Hopf algebra model of the based loop space given by $T\left({H}_{\ast }\left(M\right)\left[-1\right]\right)$. When ${C}_{\ast }$ has cyclic ${C}_{\infty }$ coalgebra structure, we describe an ${A}_{\infty }$ algebra on ${C}_{\ast }\otimes {H}_{\ast }$. 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