Homotopy algebra structures on twisted tensor products and string topology operations

Miller, Micah¹

¹Department of Mathematics, The Graduate Center at CUNY, 365 Fifth Ave, New York NY 10016, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 1163-1203

Résumé

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.

Détail

DOI : 10.2140/agt.2011.11.1163

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

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