HOMOTOPY ALGEBRA STRUCTURES ON TWISTED TENSOR PRODUCTS AND STRING TOPOLOGY OPERATIONS

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