Dan said: 7 is the lowest dimension d *known* for which the sphere S^d of d dimensions (all points at a distance of 1 from the origin in R^(d+1) has more than one inequivalent differentiable structure. (This was discovered by John Milnor in 1956, in his coyly titled paper "On manifolds homeomorphic to the 7-sphere". At that time it had been expected that each topological manifold admits a unique differentiable structure. Later work showed that S^7 has exactly 28 inequivalent (classes of) differentiable structures. Me: There is the following entry in the OEIS - any comments or updates would be welcomed. I believe further terms are known to the toplogists, but I have not been able to get hold of them. Neil %I A001676 M5197 N2261 %S A001676 1,1,1,1,1,1,28,2,8,6,992,1,3,2,16256,2,16,16 %N A001676 Number of h-cobordism classes of smooth homotopy n-spheres. %C A001676 For n not equal to 3 or 4 (and possibly for all n) this is the number of oriented diffeomorphism classes of differentiable structures on the n-sphere. %C A001676 a(3) = 1 follows now that the Poincare conjecture has been proved. %D A001676 M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963 504-537. %D A001676 S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016] %D A001676 S. O. Kochman and M. E. Mahowald, On the computation of stable stems. The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018] %D A001676 J. P. Levine, Lectures on groups of homotopy spheres. In Algebraic and geometric topology (New Brunswick, NJ, 1983), 62-95, Lecture Notes in Math., 1126, Springer, Berlin, 1985. %D A001676 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285. %D A001676 S. P. Novikov ed., Topology I, Encyc. of Math. Sci., vol. 12. %D A001676 H. Whitney, The work of John W. Milnor, pp. 48-50 of Proc. Internat. Congress Mathematicians, Stockholm, 1962. %H A001676 A. Hatcher, <a href="http://www.math.cornell.edu/~hatcher/stemfigs/stems.html">Stable Homotopy Groups of Spheres</a> %H A001676 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98). %H A001676 E. W. Weisstein, <a href="http://mathworld.wolfram.com/ExoticSphere.html">Link to a section of The World of Mathematics.</a> %Y A001676 Cf. A047680, A053381, A057617, A048648. %K A001676 nonn,hard,nice %O A001676 1,7 %A A001676 njas