Exotic Sphere
Milnor (1963) found more than one smooth structure on the 7-D Hypersphere. Generalizations have subsequently been foundin other dimensions. Using Surgery theory, it is possible to relate the number of Diffeomorphism classes ofexotic 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 bythe 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, ... (Sloane's A001676).The status of 
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. SequenceA001676/M5197in ``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.
- Almost Alternating Knot
- Almost Alternating Link
- Almost Everywhere
- Almost Integer
- Almost Perfect Number
- Almost Prime
- Alpha
- Alphabet
- Alpha Function
- Alphamagic Square
- Alphametic
- Alpha Value
- Alternate Algebra
- Alternating Algebra
- Alternating Group
- Alternating Knot
- Alternating Knot Diagram
- Alternating Link
- Alternating Permutation
- Alternating Series
- Alternating Series Test
- Alternative Link
- Altitude
- Alysoid
- Ambient Isotopy