At the Frontier of Spacetime pp 241-246 | Cite as

# Exotic Smoothness on Spheres

## Abstract

In his article [13], “Differential Topology Forty-six Years Later” published in the Notices of the AMS in 2011, John Milnor posed the following problem. Is the finite abelian group of oriented diffeomorphism classes of closed smooth homotopy spheres of dimension *n* nontrivial for all dimensions \(n>6\) with *n* different from 12 and 61? He includes a table enumerating these groups of closed smooth homotopy spheres for all \(n<64\), *n* different from 4. Nontriviality of the group of distinct exotic smoothness structures on the n-dimensional sphere provides counterexamples to the differential Poincare hypothesis in dimension *n*. We note in this abstract that this problem posed by John Milnor has a nearly complete solution, principally due to constructions of infinite 2-primary families of nontrivial elements in the stable homotopy of spheres by numerous topologists and also by the recent spectacular work of Hill et al. [7] on the non-existence of Kervaire invariant one elements in all dimensions \(2^{j}-2\) with \(j>7.\) We obtain the main theorem: The n-dimensional sphere, \(S^{n},\) admits exotic smoothness structures of 2-primary order for all dimensions \(n>61\) such that *n* is not congruent to 4 modulo 8 and also *n* not equal to 125 or 126. Moreover, for any integer *n* congruent to 4 modulo 8 of the form \(n=4p(p-1)-4\) for some odd prime \(p,S^{n}\) admits exotic differentiable structures of p-primary order.

## Keywords

Homotopy Group Smooth Structure Special Orthogonal Group Homotopy Sphere Stable Homotopy Group## Notes

### Acknowledgments

Finally, we wish to express our sincere gratitude to Professor Carl Brans for his collaboration and friendship at Loyola University. We also thank Torsten Asselmeyer-Maluga for his work on this Festschrift and his very interesting work with Carl on exotic smoothness in physics.

## References

- 1.J.C. Becker, D.H. Gottlieb, The transfer map and fiber bundles. Topology
**14**, 1–12 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 2.M. Behrens, M. Hill, M.J. Hopkins, M. Mahowald, Exotic spheres detected by topological modular forms. PreprintGoogle Scholar
- 3.E.H. Connell, A topological H-cobordism theorem for \(n \ge 5\). Ill. J. Math.
**11**, 300–309 (1967)MathSciNetzbMATHGoogle Scholar - 4.C. Daniel, Isaksen. Stable Stems. arXiv:1407.8418
- 5.M.H. Freedman, The topology of four-dimensional manifolds. J. Differ. Geom.
**17**, 357–453 (1982)MathSciNetzbMATHGoogle Scholar - 6.H. Freudenthal, Über die Klassen der Sphärenabbildungen. Comput. Math.
**5**, 299–314 (1937)MathSciNetzbMATHGoogle Scholar - 7.M.A. Hill, M.J. Hopkins, C. Douglas, Ravenel. On the non-existence of elements of Kervaire invariant one. arXiv:0908.3724
- 8.M.A. Kervaire, J.W. Milnor, Groups of homotopy spheres: I. Ann. Math. (Princeton University Press)
**77**(3), 504–537 (1963)MathSciNetCrossRefzbMATHGoogle Scholar - 9.S.O. Kochman, M.E. Mahowald, On the computation of stable stems. Contemp. Math.
**181**, 299–316 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 10.M. Mahowald. The order of the image of the J-homomorphism, in
*Proceedings of Advanced Study Inst. on Algebraic Topology*, Aarhus, vol. II, (Inst., Aarhus Univ, Mat, 1970), p. 376–384Google Scholar - 11.J.P. May, The cohomology of restricted Lie algebras and of Hopf algebras; application to the Steenrod algebra. Thesis. (The Department of Mathematics, Princeton University, 1964)Google Scholar
- 12.J.W. Milnor, On manifolds homeomorphic to the 7-sphere. Ann. Math.
**64**(2), 399–405 (1956)MathSciNetCrossRefzbMATHGoogle Scholar - 13.J.W. Milnor, Differential topology forty-six years later. Not. Am. Math. Soc.
**58**(6), 804–809 (2011)MathSciNetzbMATHGoogle Scholar - 14.E.E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. Math.
**56**, 96–114 (1952)Google Scholar - 15.J.W. Morgan, G. Tian,
*Ricci Flow and the Poincaré Conjecture*. Clay Mathematics Institute Monograph Vol. 3 (AMS and Clay Math. Institute 2007). http://www.claymath.org/library/monographs/cmim03c.pdf - 16.J.W. Morgan, F.T.-H. Fong,
*Ricci Flow and Geometrization of 3-Manifolds*. University Lecture Vol. 53 (AMS, 2010)Google Scholar - 17.M.H.A. Newman, The engulfing theorem for topological manifolds. Ann. Math.
**84**, 555–571 (1966)MathSciNetCrossRefzbMATHGoogle Scholar - 18.G. Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
- 19.D. Quillen, The Adams conjecture. Topology
**10**, 67–80 (1971)MathSciNetCrossRefzbMATHGoogle Scholar - 20.S. Smale, Generalized Poincarés conjecture in dimensions greater than four. Ann. Math.
**74**, 391–406 (1961)MathSciNetCrossRefzbMATHGoogle Scholar - 21.D.P. Sullivan, Genetics of homotopy theory and the Adams conjecture. Ann. Math.
**100**, 1–79 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 22.G. Wang, Z. Xu, On the uniqueness of the smooth structure of the 61-sphere. arXiv:1601.02184