A048648 -id:A048648 - cp's OEIS frontend

A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

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, 1, 8, 4, 1986677733776616536315084668928, 4, 1, 24, 284423744326342962334231917756416

Offset: 1

N. J. A. Sloane

For n not equal to 4 (and possibly for all n) this is the number of oriented diffeomorphism classes of differentiable structures on the n-sphere.
a(3) = 1 follows now that the Poincaré conjecture has been proved.
a(n) for n != 4 is the order of S_n, the n-th group in Tables 1 and 2 (explained in Further Details p. 807) of Milnor 2011.
The sequence is essentially given in the rightmost column of tables 1 and 2 in Isaksen, Wang & Xu (2020). It corrects some errors in earlier work. - Andrey Zabolotskiy, Nov 27 2022

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]
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]
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.
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. 48-50 of Proc. Internat. Congress Mathematicians, Stockholm, 1962.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 using data from Isaksen, Wang & Xu (2023).
Tom Copeland, The Kervaire-Milnor formula
Brady Haran and Ciprian Manolescu, The Puzzling Fourth Dimension (and exotic shapes), Numberphile video (2022).
Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine, Aug 22 2023.
A. Hatcher, Stable Homotopy Groups of Spheres
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres, PNAS, 117 (2020), 24757-24763.
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243.
M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I, Ann. of Math. (2) 77 1963 504-537.
S. S. Khare, On Abel Prize 2011 to John Willard Milnor, Math. Student, 82 (2013), 247-279. [dead link]
Alexander Kupers, Lectures on diffeomorphism groups of manifolds, Version Apr 28 2018.
J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956), 399-405.
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
G. D. Rizell, J. D. Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol. 8 (2015) 586-602
Anthony Saint-Criq, What is so exotic about dimension four?, Univ. de Toulouse (France 2022).
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

Cf. A053381, A057617, A048648, A187595, A187717, A189995, A191783, A228689, A228690, A228691, A228692.

More terms from Paul Muljadi, Mar 17 2011
Further terms from Jonathan Sondow, Jun 16 2011
The terms a(56), a(57), a(63) corrected by Andrey Zabolotskiy, Nov 27 2022

A189996 Bott periodicity: the homotopy groups of the stable orthogonal group are periodic with period 8 and repeat like [2, 2, 1, 0, 1, 1, 1, 0].

Original entry on oeis.org

2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0

Offset: 0

Jonathan Sondow, Jun 17 2011

Bott proved that the n-th homotopy group of the stable orthogonal group is Z/(a(n)*Z), where Z is the integers and Z/(0*Z), Z/(1*Z), Z/(2*Z) are the cyclic groups of order infinity, 1, 2, respectively. For details, see the Wikipedia orthogonal group link.

For references and additional links, see the Wikipedia Bott periodicity link.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Bott Periodicity Theorem
Wikipedia, Bott periodicity
Wikipedia, Orthogonal group
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A048648.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{2, 2, 1, 0, 1, 1, 1, 0},104] (* Ray Chandler, Aug 25 2015 *)
PadRight[{},120,{2,2,1,0,1,1,1,0}] (* Harvey P. Dale, Jun 13 2017 *)

a(n)=[2, 2, 1, 0, 1, 1, 1, 0][n%8+1] \\ Charles R Greathouse IV, Jul 13 2016

Vec((2 + 2*x + x^2 + x^4 + x^5 + x^6) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^90)) \\ Colin Barker, Nov 02 2019

a(n) = 2, 2, 1, 0, 1, 1, 1, 0 if n == 0, 1, 2, 3, 4, 5, 6, 7 (mod 8), respectively.
From Colin Barker, Nov 02 2019: (Start)
G.f.: (2 + 2*x + x^2 + x^4 + x^5 + x^6) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n-8) for n>7.
(End)

A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

Original entry on oeis.org

1, 2, 1, 24, 1, 1, 1, 240, 2, 2, 1, 504, 1, 1, 1, 480, 2, 2, 1, 264, 1, 1, 1, 65520, 2, 2, 1, 24, 1, 1, 1, 16320, 2, 2, 1, 28728, 1, 1, 1, 13200, 2, 2, 1, 552, 1, 1, 1, 131040, 2, 2, 1, 24, 1, 1, 1, 6960, 2, 2, 1, 171864, 1, 1, 1, 32640, 2, 2, 1, 24, 1, 1, 1

Offset: 0

Tom Harris, Jun 12 2021

Im(J) is a finite cyclic subgroup of Pi_n^S and has known order a(n) calculated by Adams using the Adams conjecture, subsequently proven by Quillen. When n is 3 or 7 mod 8 the value a(n) is related to the Bernoulli numbers; the other values of a(n) are 8-periodic (after an exceptional n=0).

D. Ravenel, Complex cobordism and stable homotopy groups of spheres (2ed), AMS Chelsea Publishing, (2003), ISBN: 978-0-8218-2967-7.

J.F. Adams, On the groups J(x), IV, Topology 5 (1966), 21-71.
D.G. Quillen, The Adams conjecture, Topology 10 (1971), 1-10.

Cf. A006863, A079612. Divides A048648.

from sympy import bernoulli
def a(n):
    if n == 0:
        return 1
    n_ = n % 8
    d = {0:2, 1:2, 2:1, 4:1, 5:1, 6:1}
    if n_ in [3, 7]:
        k = (n+1)//4
        return (bernoulli(2*k)/(4*k)).denominator
    else:
        return d[n_]

a(n) is:
                 2 if n = 0 or 1 mod 8 (except a(0) = 1)
                 1 if n = 2, 4, 5 or 6 mod 8
  A006863((n+1)/4) if n = 3 or 7 mod 8.
(A006863(k) = denominator of B_2k/4k, where B_m are the Bernoulli numbers.)

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A122962 Duplicate of A048648.

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A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

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A189996 Bott periodicity: the homotopy groups of the stable orthogonal group are periodic with period 8 and repeat like [2, 2, 1, 0, 1, 1, 1, 0].

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A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

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