Exotic Sphere

Milnor (1963) found more than one smooth structure on the 7-D 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 $N(d)$ of distinct (up to Diffeomorphism) Differential Structures on spheres indexed by the Dimension $d$ of the sphere. For $d=1$, 2, ..., assuming the Poincaré Conjecture, they are 1, 1, 1, $\geq 2$, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ... (Sloane's A001676). The status of $d=4$ is still unresolved: at least one exotic structure exists, but it is not known if others do as well.

The only exotic Euclidean spaces are a Continuum of Exotic R4 structures.

See also Exotic R4, Hypersphere

References

Kervaire, M. A. and Milnor, J. W. ``Groups of Homotopy Spheres: I.'' Ann. Math. 77, 504-537, 1963.

Kosinski, A. A. §X.6 in Differential Manifolds. Boston, MA: Academic Press, 1992.

Milnor, J. ``Topological Manifolds and Smooth Manifolds.'' 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, 1997.

Novikov, S. P. (Ed.). Topology I. New York: Springer-Verlag, 1996.

Sloane, N. J. A. Sequence A001676/M5197 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.