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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.

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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!

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  • $\begingroup$ Sorry that this is somewhat a tangent, but is there a consensus among experts about the truth/falsity of smooth 4D Poincaré? $\endgroup$ Aug 18 at 15:19
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    $\begingroup$ There is not, though recently a collection of potential counterexamples have been popping up and simultaneously being proven standard. A little anecdote that I was present for: During one of the Harvard gauge theory seminars Cliff Taubes commented "I read on Wikipedia that the experts think the smooth Poincaré conjecture is true", and then everyone in the audience laughed; indeed most of the experts were in that audience that day (I think Danny Ruberman was there). $\endgroup$ Aug 18 at 15:46

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