A classic yet delicate fact of Morse theory states that the unstable manifolds of a Morse-Smale gradient-flow on a closed manifold M are the open cells of a CW-decomposition of M. I will describe a self-contained proof by Abbondandolo and myself. The key tool is a "system of invariant stable foliations", which is analogous to the object introduced by Palis and Smale in their proof of structural stability of Morse Smale diffeomorphisms and flows, but with finer regularity and geometric properties.
[Stable foliations and CW structure induced by a Morse-Smale gradient flow, A.Abbondandolo,P.Majer] https://www.worldscientific.com/doi/10.1142/S1793525321500527
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006