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.

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A. A. §X.6 in Differential
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"Lectures on Groups of Homotopy Spheres." In Algebraic and Geometric
Topology (New Brunswick, NJ, 1983). Berlin: Springer-Verlag, pp. 62-95,
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J. "Topological Manifolds and Smooth Manifolds." In Proc. Internat.
Congr. Mathematicians (Stockholm, 1962). Djursholm: Inst. Mittag-Leffler, pp. 132-138,
1963.Milnor, J. W. and Stasheff, J. D. Characteristic
Classes. Princeton, NJ: Princeton University Press, 1973.Monastyrsky,
M. Modern
Mathematics in the Light of the Fields Medals. Wellesley, MA: A K Peters,
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I. New York: Springer-Verlag, 1996.Scorpan, A. The
Wild World of 4-Manifolds. Providence, RI: Amer. Math. Soc., 2005.Sloane,
N. J. A. Sequence A001676/M5197
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