A001676 -id:A001676 - cp's OEIS frontend

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.

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.

A187595 Bernoulli-Kervaire-Milnor numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 28, 1, 2, 1, 992, 1, 1, 1, 8128, 1, 2, 1, 261632, 1, 2, 1, 1448424448, 1, 2, 1, 67100672, 1, 1, 1, 1941802827776, 1, 2, 1, 753623571759104, 1, 2, 1, 23998307331473408, 1, 2, 1, 341653284209033216, 1, 2, 1, 8316321134799694594048, 1, 2, 1, 740764429532373450752, 1, 2, 1, 30559446583872811817762816, 1, 2, 1, 496669433444154134078771167232, 1, 1, 1, 17776484020396435145889494859776

Offset: 1

Paul Muljadi, Mar 11 2011

nonn

|bP_(n+1)| in 1961 Kervaire-Milnor theorem.
|b_n| in Milnor 2011 Table 2 (explained in Further Details p. 807). - Jonathan Sondow, Jun 16 2011
Named after the Swiss mathematician Jacob Bernoulli (1655-1705), the French mathematician Michel André Kervaire (1927-2007) and the American mathematician John Willard Milnor (b. 1931). - Amiram Eldar, Jun 22 2021

Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I, December 1961.
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I, Annals of Mathematics, Second Series, Vol. 77, No. 3 (1963), pp. 504-537; alternative link.
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc., Vol. 58, No. 6 (2011), pp. 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
Georgios Dimitroglou Rizell and Jonathan David Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol., Vol. 8, No. 2 (2015), pp. 586-602.
Wikipedia, Exotic sphere.

Cf. A001676.

More terms from Jonathan Sondow, Jun 16 2011.

A057617 Order of subgroup bP_{m+1} of group Theta_m of h-cobordism classes of smooth homotopy m-spheres defined by those homotopy m-spheres which bound parallelizable (m+1)-manifolds, where m = 2n+1.

Original entry on oeis.org

1, 1, 1, 28, 2, 992, 1, 8128, 2, 130816

Offset: 0

N. J. A. Sloane, Feb 03 2002

nonn

a(1) = 1 because the Poincaré conjecture is true.

M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963 504-537.
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.

John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.

Cf. A001676.

a(0)-a(1) from Andrey Zabolotskiy, Feb 02 2018

A191783 Numbers k such that the topological k-sphere has a unique differentiable structure up to diffeomorphism.

Original entry on oeis.org

1, 2, 3, 5, 6, 12, 56, 61

Offset: 1

N. J. A. Sloane, Jun 25 2011

Whether 4 is a term is an open question. - Andrey Zabolotskiy, Feb 02 2018
Except (possibly) for k=4, these are the numbers k such that A001676(k)=1. - Jeppe Stig Nielsen, May 22 2019
The list in Milnor's paper (p. 807, A358290) does not include the term 56 because it was only discovered later that it is actually a term of this list, see Wang & Xu, Theorem 1.14. - Andrey Zabolotskiy, Nov 29 2022
Dimensions without exotic spheres. - Charles R Greathouse IV, Dec 22 2022

Brady Haran and Ciprian Manolescu, The Puzzling Fourth Dimension (and exotic shapes), Numberphile video (2022).
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
Guozhen Wang and Zhouli Xu, The triviality of the 61-stem in the stable homotopy groups of spheres, Annals of Mathematics, 186 (2017), 501-580; arXiv:1601.02184 [math.AT], 2016-2017.

Cf. A001676, A358290.

Definition rewritten by Jeppe Stig Nielsen, May 22 2019

The term a(7) = 56 inserted by Andrey Zabolotskiy, Nov 27 2022

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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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A187595 Bernoulli-Kervaire-Milnor numbers.

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A057617 Order of subgroup bP_{m+1} of group Theta_m of h-cobordism classes of smooth homotopy m-spheres defined by those homotopy m-spheres which bound parallelizable (m+1)-manifolds, where m = 2n+1.

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Extensions

A191783 Numbers k such that the topological k-sphere has a unique differentiable structure up to diffeomorphism.

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Author

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Extensions