In [[KM63]](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/kervmiln.pdf), Kervaire and Milnor introduced the *group of homotopy spheres*. Its elements are h-cobordism classes of smooth homotopy $n$-spheres under the summation induced by connected sum. Further, the trivial element is $S^n$ and this group is denoted by $\Theta^n$.


They proved that $\Theta^n$ is finite unless $n=3$, in particular $\Theta^4$ is trivial.


This should be an ambiguous question but I wonder this provides a positive clue for the smooth Poincaré conjecture in dimension 4.



Kervaire-Milnor group of homotopy spheres and smooth Poincaré conjecture
That a homotopy 4-sphere is h-cobordant to $S^4$ is in principle a step towards proving the 4-dimensional Poincaré conjecture. But it's known from Donaldson's work that the h-cobordism theorem is false for simply connected closed $4$-manifolds. Indeed the step that fails is in cancelling handles that homologically cancel, and that issue would come up in trying to prove the Poincaré conjecture starting from an h-cobordism.


So unless there's something special about trivializing an h-cobordism between homotopy spheres that doesn't hold for slightly more complicated $4$-manifolds, I'd say that this fact isn't much of a clue.


Just an opinion, of course!


