# Kervaire-Milnor group of homotopy spheres and smooth Poincaré conjecture

In [KM63], 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.

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.