Homotopy algebra structures on twisted tensor products and string topology operations

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 $M\to M\times M$ 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.