Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations

الأصل · EN

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 (non-minimal) A∞ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal G-bundles.