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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

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 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

References

• 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.

Links

• 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

Crossrefs

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

Extensions

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