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 the orders of groups of oriented homotopy
spheres. The numbers 
of oriented smooth structures on spheres up to
orientation-preserving diffeomorphism, including
the standard smooth structure, begin for dimension 
, 2, ... with 1, 1, 1, 
, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ...
(OEIS A001676). The value 1 at 
in OEIS A001676 counts
h-cobordism classes and does not resolve the
smooth classification. 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 themselves.
Galaz-García (2026) announced positively curved exotic spheres in infinitely many dimensions. The reported construction gives at
least one exotic sphere with strictly positive sectional
curvature in every dimension 
with 
, except the congruence class 
. A computer calculation also covers that congruence
class below 
.
This is an existence statement, not a result for every
smooth structure. The curvature conclusion depends
on recent geometric constructions and a gluing theorem
retained as a hypothesis in the accompanying formalization.
Galaz-García (2026) reports AI assistance with mathematical exploration, programming,
and review.
-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.