Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations

TL;DR

This paper develops a method to construct $A__$ algebra and coalgebra structures on twisted tensor products, providing models for string topology operations and loop spaces using homotopy algebra techniques.

Contribution

It extends Brown's twisted tensor product construction to $A__$ structures and applies it to model chains on free loop spaces and string topology operations.

Findings

Constructs $A__$ coalgebra on twisted tensor products.

Provides an explicit $A__$ algebra model of the string topology loop product.

Describes a representation of the loop product in principal $G$-bundles.

Abstract

Given a $C_\infty$ coalgebra $C_*$, a strict dg Hopf algebra $H_*$, and a twisting cochain $\tau:C_* \rightarrow H_*$ such that $Im(\tau) \subset Prim(H_*)$, 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 $LM$ based on the $C_\infty$ coalgebra structure of $H_*(M)$ induced by the diagonal map $M \rightarrow M \times M$ 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 (non-minimal) $A_\infty$ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal $G$-bundles.