Higher String Topology on General Spaces

In [2], Chas and Sullivan considered the free loop space LM = Maps(S,M) for a smooth orientable compact manifold M . They used geometric methods to show that H∗LM has, among other things, the structure of a Batalin-Vilkovisky algebra. More precisely, C∗LM has an action by the operad of chain complexes C∗D2, where D2 is the framed 2-dimensional little disk operad. Each element of D2(n) is an ordered configuration of n disjoint 2-dimensional little disks d1, . . . dn inside the 2-dimensional disk D, together with a linear parametrization λi : S 1 → ci, where ci is the boundary of di, for each i. Also, for such an M , there is a chain homotopy equivalence