A228689 -id:A228689 - 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

A189995 The order b_{4n-1} of the cyclic group S_{4n-1}^{bp} of oriented diffeomorphism classes of smooth homotopy (4n-1)-spheres that bound parallelizable manifolds, for n > 1.

Original entry on oeis.org

28, 992, 8128, 261632, 1448424448, 67100672, 1941802827776, 753623571759104, 23998307331473408, 341653284209033216, 8316321134799694594048, 740764429532373450752, 30559446583872811817762816, 496669433444154134078771167232, 17776484020396435145889494859776, 11188223110510348416175908585472

Offset: 2

Jonathan Sondow, Jun 15 2011

nonn

For a(n), Milnor 2011 Theorem 5 gives the formula
2^(2*n-2)*(2^(2*n-1)-1)*numerator(4*bernoulli(n)/n)
where bernoulli(n) = abs(Bernoulli(2*n)).
See A001676 for additional comments, references, and links.

			a(2) = 2^2 * (2^3 - 1) * abs(numerator(4 * Bernoulli(4)/2)) = 4 * 7 * abs(numerator(2 * (-1/30))) = 28

J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.

G. C. Greubel, Table of n, a(n) for n = 2..235
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809 (see Theorem 5 and Table 3).
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.

Cf. A001676, A047680, A057617, A187595, A228689.

[2^(2*n-2)*(2^(2*n-1)-1)*Abs(Numerator(4*Bernoulli(2*n)/n)): n in [2..30]]; // G. C. Greubel, Jan 11 2018

Table[2^(2*n-2)*(2^(2*n-1)-1)*Abs[Numerator[4*BernoulliB[2*n]/n]],{n,2,17}]

a(n) = 2^(2*n - 2) * (2^(2*n - 1) - 1) * abs(numerator(4*Bernoulli(2*n)/n)).

a(n) = A187595(4*n-1) for n > 1.

A228690 a(n) = A001676(4*n).

Original entry on oeis.org

1, 2, 1, 2, 24, 2, 2, 8, 6, 192, 8, 32, 24, 1, 4, 128, 24, 384, 80, 256

Offset: 1

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228691, A228692.

Term a(14) corrected, a(16)-a(20) added by Andrey Zabolotskiy, Nov 27 2022

A228691 a(n) = A001676(4*n + 1).

Original entry on oeis.org

1, 1, 8, 3, 16, 8, 4, 3, 32, 24, 32, 11520, 12, 32, 8, 1, 6144, 32, 128, 256, 9216

Offset: 0

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228690, A228692.

Term a(14) corrected, a(16)-a(20) added by Andrey Zabolotskiy, Nov 27 2022

A228692 a(n) = A001676(4n+2).

Original entry on oeis.org

1, 1, 6, 2, 16, 4, 12, 3, 32, 120, 96, 48, 24, 8, 4, 24, 512, 512, 384, 384, 10752

Offset: 0

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228690, A228691.

More terms from the b-file at A001676 added by Amiram Eldar, May 03 2024

A228038 Dimensions in which nonzero Arf-Kervaire invariants exist.

Original entry on oeis.org

2, 6, 14, 30, 62, 126

Offset: 2

Jonathan Sondow, Sep 01 2013

Hill, Hopkins, and Ravenel (2009) proved that nonzero Arf-Kervaire invariants exist only in dimensions 2^n - 2 for n = 2, 3, 4, 5, 6, and possibly 7, that is, in dimensions 2, 6, 14, 30, 62 and possibly 126.

The preprint by Lin, Wang, and Xu asserts that nonzero Arf-Kervaire invariants exist in dimension 126.

J. Baez, ‘Kervaire Invariant One Problem’ Solved, The n-Category Café (blog), April 2009.
M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, arXiv:0908.3724 [math.AT], 2009-2015.
Erica Klarreich, Dimension 126 Contains Strangely Twisted Shapes, Mathematicians Prove, Quanta Magazine, May 2025.
Weinan Lin, Guozhen Wang, and Zhouli Xu, On the Last Kervaire Invariant Problem, arXiv:2412.10879 [math.AT], 2024-2025.
V. P. Snaith, A history of the Arf-Kervaire invariant problem, Notices Amer. Math. Soc., 60 (No. 8, 2013), 1040-1047.
Wikipedia, Kervaire invariant

Cf. A228689, A000918.

a(n) = 2^n - 2 for n = 2, 3, 4, 5, 6, 7.

a(7) from David Radcliffe, May 09 2025

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

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A189995 The order b_{4n-1} of the cyclic group S_{4n-1}^{bp} of oriented diffeomorphism classes of smooth homotopy (4n-1)-spheres that bound parallelizable manifolds, for n > 1.

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A228690 a(n) = A001676(4*n).

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A228691 a(n) = A001676(4*n + 1).

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A228692 a(n) = A001676(4n+2).

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A228038 Dimensions in which nonzero Arf-Kervaire invariants exist.

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