Invariants of 3-manifolds via Morse homotopy

K. Fukaya defined an invariant of a 3-manifold equipped with an acyclic flat bundle via a Morse theoretic method. Fukaya's invariant is conjectured to coincide with the 2-loop part of perturbative Chern--Simons theory through the Witten deformation. We define, as a higher-order generalization of the construction of Fukaya, an invariant of homology 3-spheres by the numbers of gradient graph flows on a homology 3-sphere.
We conjecture that our construction could be extended to closed 3-manifolds with the first Betti number 1.