An exotic sphere is a smooth manifold that is homeomorphic but not diffeomorphic to a hypersphere. Milnor (1956) found more than one smooth structure on the seven-dimensional hypersphere. Generalizations have subsequently been found in other dimensions. Using surgery theory, it is possible to relate the number of diffeomorphism classes of exotic spheres to higher homotopy groups of spheres (Kosinski 1992).
Kervaire and Milnor (1963) computed a list of the number 
of distinct (up to diffeomorphism)
differential structures on spheres indexed by the dimension 
of the sphere. For 
, 2, ..., assuming the Poincaré
conjecture, they are 1, 1, 1, 
, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ...
(OEIS A001676). The status of 
is still unresolved, and it is not known whether there is
1, more than 1, or infinitely many smooth structures on the 4-sphere (Scorpan 2005).
The claim that there is exactly one is known as the smooth Poincaré conjecture
for 
.
The only exotic Euclidean spaces are a continuum of exotic R4 structures.
He et al. (2026) announced that every seven-dimensional homotopy sphere admits a Riemannian metric with strictly positive sectional curvature. Their construction covers all 28 oriented diffeomorphism classes. If verified, positive sectional curvature therefore excludes none of the smooth structures on the seven-dimensional hypersphere.
The authors credit ChatGPT 6 Astra and Claude Pro with assisting their exploration of some proof strategies and calculations, and state that they performed the verification and exposition themselves.
-Sphere." Ann. Math. 64, 399-405, 1956.Milnor,
J. "Topological Manifolds and Smooth Manifolds." In Proc. Internat.
Congr. Mathematicians (Stockholm, 1962). Djursholm, Sweden: Inst. Mittag-Leffler,
pp. 132-138, 1963.Milnor, J. W. and Stasheff, J. D.