

A001676


Number of hcobordism classes of smooth homotopy nspheres.
(Formerly M5197 N2261)


12



1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4, 69524373504, 2, 4, 12, 67100672, 2, 3, 3, 7767211311104, 8, 32, 32, 3014494287036416, 6, 24, 120, 2303837503821447168, 192, 32, 96, 341653284209033216, 8, 11520, 48, 798366828940770681028608, 32, 12, 24, 11852230872517975212032, 24, 32, 8, 91678339751618435453288448, 2, 16, 4, 1986677733776616536315084668928, 4, 1, 24, 142211872163171481167115958878208
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,7


COMMENTS

For n not equal to 4 (and possibly for all n) this is the number of oriented diffeomorphism classes of differentiable structures on the nsphere.
a(3) = 1 follows now that the PoincarĂ© conjecture has been proved.
a(n) for n != 4 is the order of S_n, the nth group in Tables 1 and 2 (explained in Further Details p. 807) of Milnor 2011.


REFERENCES

M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963 504537.
S. S. Khare, On Abel Prize 2011 to John Willard Milnor, Math. Student, 82 (2013), 247279.
S. O. Kochman, Stable homotopy groups of spheres. A computerassisted approach. Lecture Notes in Mathematics, 1423. SpringerVerlag, Berlin, 1990. 330 pp. ISBN: 3540524681. [Math. Rev. 91j:55016]
S. O. Kochman and M. E. Mahowald, On the computation of stable stems. The Cech Centennial (Boston, MA, 1993), 299316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
J. P. Levine, Lectures on groups of homotopy spheres. In Algebraic and geometric topology (New Brunswick, NJ, 1983), 6295, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
J. W. Milnor, On manifolds homeomorphic to the 7sphere, Ann. of Math. 64 (1956), 399405.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
S. P. Novikov ed., Topology I, Encyc. of Math. Sci., vol. 12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. Whitney, The work of John W. Milnor, pp. 4850 of Proc. Internat. Congress Mathematicians, Stockholm, 1962.


LINKS

Table of n, a(n) for n=1..63.
A. Hatcher, Stable Homotopy Groups of Spheres
John W. Milnor, Differential Topology Fortysix Years Later, Notices Amer. Math. Soc. 58 (2011), 804809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Exotic Sphere.
Wikipedia, Exotic sphere


CROSSREFS

Cf. A047680, A053381, A057617, A048648, A187595, A187717, A189995, A228689A228692.
Sequence in context: A257835 A040780 A040781 * A040782 A040783 A057617
Adjacent sequences: A001673 A001674 A001675 * A001677 A001678 A001679


KEYWORD

nonn,hard,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Paul Muljadi, Mar 17 2011
Further terms from Jonathan Sondow, Jun 16 2011


STATUS

approved



