## Algebraic & Geometric Topology

### Homotopy algebra structures on twisted tensor products and string topology operations

Micah Miller

#### Abstract

Given a $C∞$ coalgebra $C∗$, a strict dg Hopf algebra $H∗$ and a twisting cochain $τ:C∗→H∗$ such that $Im(τ)⊂ Prim(H∗)$, we describe a procedure for obtaining an $A∞$ coalgebra on $C∗⊗H∗$. This is an extension of Brown’s work on twisted tensor products. We apply this procedure to obtain an $A∞$ coalgebra model of the chains on the free loop space $LM$ based on the $C∞$ coalgebra structure of $H∗(M)$ induced by the diagonal map $M→M×M$ and the Hopf algebra model of the based loop space given by $T(H∗(M)[−1])$. When $C∗$ has cyclic $C∞$ coalgebra structure, we describe an $A∞$ algebra on $C∗⊗H∗$. This is used to give an explicit (nonminimal) $A∞$ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal $G$–bundles.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 1163-1203.

Dates
Revised: 31 January 2011
Accepted: 4 February 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715223

Digital Object Identifier
doi:10.2140/agt.2011.11.1163

Mathematical Reviews number (MathSciNet)
MR2792377

Zentralblatt MATH identifier
1220.55005

#### Citation

Miller, Micah. Homotopy algebra structures on twisted tensor products and string topology operations. Algebr. Geom. Topol. 11 (2011), no. 2, 1163--1203. doi:10.2140/agt.2011.11.1163. https://projecteuclid.org/euclid.agt/1513715223

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