A-infinity structures in topology

From Talbot

There are two main things: A_infty structures underlying the cohomology of a manifold, using Morse theory (Fukaya-Oh or Cohen-Norbury, hopefully trying to be a little more precise by constructing an actual A_\infty structure on Morse cohomology?).

The other classical example is the homology (not cohomology) of the loop space. One can actually generalize this to open string theory (space of paths between given manifolds, with the string product). That generalization is nontrivial due to issues with chain level intersection products. Morse theory can actually help here as well. Maybe someone feels like tackling it?