Applications of string topology structure

Hossein Abbaspour gave an interesting connection between 3-manifold topology and the string topology algebraic structure in arXiv:0310112. The map $M to LM$ given by sending a point $x$ to the constant loop at $x$ allows one to split

$mathbb{H}_*(LM)$ as $H_*(M) oplus A_M$.

He showed essentially that the restriction of
the string product to the $A_M$ summand is nontrivial if and only if $M$ is hyperbolic. There are some technical details in the statements in his paper, but it was written pre-Perelman and I believe the statements can be made a bit more elegant in light of the Geometrization Theorem.

Philosophically, Sullivan has said that he his goal in inventing string topology was to try to find new invariants of smooth structures on manifolds. His original idea was that if you have to use the smooth structure to smoothly put chains into transversal positions to intersect them then you might hope that the answer will depend on the smooth structure. Unfortunately, we now know that the string topology BV algebra depends only on the underlying homotopy type of the manifold (there are now quite a few different proofs of various parts of this statement).

The string topology BV algebra is only a piece of a potentially much richer algebraic structure. Roughly speaking, $mathbb{H}_*(LM)$ is a homological conformal field theory. This was believed to be true for quite some time but it took a while before it was finally produced by Veronique Godin arxiv:0711.4859. She constructed an action of the PROP made from the homology of moduli spaces of Riemann surfaces with boundary. Restricting this action to pairs of pants recovers the original Chas-Sullivan structure.

Unfortunately, for degree reasons, nearly all of the higher operations vanish. In particular, any operation given by a class in the Harer stable range of the homology of the moduli space must act by zero. Hirotaka Tamanoi has a paper that spells out the details, but it is nothing deep.

Furthermore, it seems that the higher operations are homotopy invariant as well. For instance Lurie gets this as a corollary of his work on the classification of topological field theories.

Last I heard, Sullivan, ever the optimist, believes that there is still hope for string topology to detect smooth structures. He says that one should be able to extend from the moduli spaces of Riemann surfaces to a certain piece of the boundary of the Deligne-Mumford compactification. I've heard that the partial compactification here is meant to be that one allows nodes to form, but only so long as the nodes collectively do not separate the incoming boundary components from the outgoing boundary. Sullivan now has some reasons to hope that operations coming from homology classes related to the boundary of these moduli spaces might see some information about the underlying smooth structure of the manifold.