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Homotopy algebra structures on twisted tensor products and string topology operations

Micah Miller

Algebraic & Geometric Topology 11 (2011) 1163–1203

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.

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
Received: 14 June 2010
Revised: 31 January 2011
Accepted: 4 February 2011
Published: 11 April 2011
Micah Miller
Department of Mathematics
The Graduate Center at CUNY
365 Fifth Ave
New York NY 10016