TITLE: Applications of string topology structure
QUESTION [26 upvotes]: Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure includes a product which combines the intersection product and Pontryagin product and a BV operater $\Delta: \mathbb{H}_*(LM) \to \mathbb{H}_{*+1}(LM)$.
I was wondering about the applications of this structure. Has it even been used to prove theorems in other parts of mathematics? A more concrete question is the following: Usually, considering a more complicated structure on topological invariants of a space allows you to prove certain non-existince results. For example, the cup product in cohomology allows you to distinguish between $ S^2 \vee S^1 \vee S^1 $  and 
$T^2$. Is there an example of this type for string topology?

REPLY [8 votes]: As a shameless plug, I may say that in my thesis we do show that string topology, interpreted in a broader context, is NOT a homotopy invariant. What we do is the following : instead of looking at loops in $M$ we think of them as arcs in $M\times M$ with its boundary in the diagonal $M$ that sits inside $M\times M$. Now we look at the space of such arcs $\mathcal{S}(M)$, which, when they intersect the diagonal at intermediate stages, do so transversely. One can then define a suitable coalgebra structure which is NOT a homotopy invariant. In particular, this structure distinguishes the Lens spaces $L(7,1)$ from $L(7,2)$, which are homotopy equivalent but NOT homeomorphic. 
Of course, this new structure is not related to the loop product or the BV operator as per the question asked. Moreover, this structure is defined on a much smaller space then $LM$. However, if you take the point of view that string topology is broadly the study of loops in a manifold then this is a new and interesting algebraic structure.

REPLY [2 votes]: The string topology of a manifold is isomorphic to the Hamiltonian Floer Homology of the cotangent bundle of the manifold.