Algebraic & Geometric Topology

Homotopy algebra structures on twisted tensor products and string topology operations

Micah Miller

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Given a C coalgebra C, a strict dg Hopf algebra H and a twisting cochain τ:CH such that Im(τ) Prim(H), we describe a procedure for obtaining an A coalgebra on CH. 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 MM×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 CH. 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

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

Received: 14 June 2010
Revised: 31 January 2011
Accepted: 4 February 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P35: Loop spaces 55R99: None of the above, but in this section 57N65: Algebraic topology of manifolds 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx] 57M99: None of the above, but in this section
Secondary: 55Q33 55Q32

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


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.

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