|
%I M5197 N2261 #101 Sep 14 2023 00:47:50
%S 1,1,1,1,1,1,28,2,8,6,992,1,3,2,16256,2,16,16,523264,24,8,4,
%T 69524373504,2,4,12,67100672,2,3,3,7767211311104,8,32,32,
%U 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
%N Number of h-cobordism classes of smooth homotopy n-spheres.
%C 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.
%C a(3) = 1 follows now that the Poincaré conjecture has been proved.
%C 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.
%C 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
%D 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]
%D 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]
%D 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.
%D J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
%D S. P. Novikov ed., Topology I, Encyc. of Math. Sci., vol. 12.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D H. Whitney, The work of John W. Milnor, pp. 48-50 of Proc. Internat. Congress Mathematicians, Stockholm, 1962.
%H Andrey Zabolotskiy, <a href="/A001676/b001676.txt">Table of n, a(n) for n = 1..83</a> using data from Isaksen, Wang & Xu (2023).
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/10/04/the-kervaire-milnor-formula/">The Kervaire-Milnor formula</a>
%H Brady Haran and Ciprian Manolescu, <a href="https://www.youtube.com/watch?v=CVOr7f_VALc">The Puzzling Fourth Dimension (and exotic shapes)</a>, Numberphile video (2022).
%H Kevin Hartnett, <a href="https://www.quantamagazine.org/an-old-conjecture-falls-making-spheres-a-lot-more-complicated-20230822/">An Old Conjecture Falls, Making Spheres a Lot More Complicated</a>, Quanta Magazine, Aug 22 2023.
%H A. Hatcher, <a href="http://www.math.cornell.edu/~hatcher/stemfigs/stems.html">Stable Homotopy Groups of Spheres</a>
%H Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, <a href="https://doi.org/10.1073/pnas.2012335117">Stable homotopy groups of spheres</a>, PNAS, 117 (2020), 24757-24763.
%H Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, <a href="https://doi.org/10.1007/s10240-023-00139-1">Stable homotopy groups of spheres: from dimension 0 to 90</a>, Publications mathématiques de l'IHÉS, 137 (2023), 107-243.
%H M. A. Kervaire and J. W. Milnor, <a href="http://www.jstor.org/stable/1970128">Groups of homotopy spheres: I</a>, Ann. of Math. (2) 77 1963 504-537.
%H S. S. Khare, <a href="http://www.indianmathsociety.org.in/mathstudent2013.pdf">On Abel Prize 2011 to John Willard Milnor</a>, Math. Student, 82 (2013), 247-279. [dead link]
%H Alexander Kupers, <a href="http://www.math.harvard.edu/~kupers/teaching/272x/book.pdf">Lectures on diffeomorphism groups of manifolds</a>, Version Apr 28 2018.
%H J. W. Milnor, <a href="http://www.jstor.org/stable/1969983">On manifolds homeomorphic to the 7-sphere</a>, Ann. of Math. 64 (1956), 399-405.
%H John W. Milnor, <a href="http://www.ams.org/notices/201106/rtx110600804p.pdf">Differential Topology Forty-six Years Later</a>, Notices Amer. Math. Soc. 58 (2011), 804-809.
%H John W. Milnor, <a href="https://www.youtube.com/watch?v=SIZd_xBiRS0">Spheres</a>, Abel Prize lecture (video), 2011.
%H G. D. Rizell, J. D. Evans, <a href="https://doi.org/10.1112/jtopol/jtv007">Exotic spheres and the topology of symplectomorphism groups</a>, J. Topol. 8 (2015) 586-602
%H Anthony Saint-Criq, <a href="https://www.math.univ-toulouse.fr/~asaintcr/resources/exotic_talk.pdf">What is so exotic about dimension four?</a>, Univ. de Toulouse (France 2022).
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExoticSphere.html">Exotic Sphere.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exotic_sphere">Exotic sphere</a>
%Y Cf. A053381, A057617, A048648, A187595, A187717, A189995, A191783, A228689, A228690, A228691, A228692.
%K nonn,hard,nice
%O 1,7
%A _N. J. A. Sloane_
%E More terms from _Paul Muljadi_, Mar 17 2011
%E Further terms from _Jonathan Sondow_, Jun 16 2011
%E The terms a(56), a(57), a(63) corrected by _Andrey Zabolotskiy_, Nov 27 2022
|
