Exotic Smoothness on Spheres

Abstract

In his article [13], “Differential Topology Forty-six Years Later” published in the Notices of the AMS in 2011, John Milnor posed the following problem. Is the finite abelian group of oriented diffeomorphism classes of closed smooth homotopy spheres of dimension n nontrivial for all dimensions $n>6$ with n different from 12 and 61? He includes a table enumerating these groups of closed smooth homotopy spheres for all $n<64$, n different from 4. Nontriviality of the group of distinct exotic smoothness structures on the n-dimensional sphere provides counterexamples to the differential Poincare hypothesis in dimension n. We note in this abstract that this problem posed by John Milnor has a nearly complete solution, principally due to constructions of infinite 2-primary families of nontrivial elements in the stable homotopy of spheres by numerous topologists and also by the recent spectacular work of Hill et al. [7] on the non-existence of Kervaire invariant one elements in all dimensions $2^{j}-2$ with $j>7.$ We obtain the main theorem: The n-dimensional sphere, $S^{n},$ admits exotic smoothness structures of 2-primary order for all dimensions $n>61$ such that n is not congruent to 4 modulo 8 and also n not equal to 125 or 126. Moreover, for any integer n congruent to 4 modulo 8 of the form $n=4p(p-1)-4$ for some odd prime $p,S^{n}$ admits exotic differentiable structures of p-primary order.