Exotic Smoothness, Physics and Related Topics

  • Jan SładkowskiEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 183)


In 1854 Riemann, the father of differential geometry, suggested that the geometry of space may be more than just a mathematical tool defining a stage for physical phenomena, and may in fact have profound physical meaning in its own right. Since then various assumptions about the spacetime structure have been put forward. But to what extent the choice of mathematical model for spacetime has important physical significance? With the advent of general relativity physicists began to think of the spacetime in terms a differential manifolds. In this short essay we will discuss to what extent the structure of spacetime can be determined (modelled) and the possible role of differential calculus in the due process. The counterintuitive discovery of exotic four dimensional Euclidean spaces following from the work of Freedman and Donaldson surprised mathematicians. Later, it has been shown that exotic smooth structures are especially abundant in dimension four—the dimension of the physical spacetime. These facts spurred research into possible the physical role of exotic smoothness, an interesting but not an easy task, as we will show.


Topological Space Dirac Operator Compact Space Hausdorff Space Spacetime Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A.V. Arhangelskii, Topologiceskye Prostranstva Funkcii (in Russian) (Moscow Univ. Press, Moscow, 1989)Google Scholar
  2. 2.
    A.V. Arhangelskii (ed.), General Topology III (Springer, Berlin, 1995)Google Scholar
  3. 3.
    T. Asselmeyer-Maluga, J. Król, Adv. High Energy Phys. Article Number: 867460 (2014)Google Scholar
  4. 4.
    T. Asselmeyer-Maluga, J. Król, Mod. Phys. Lett. 28 Article Number: 1350158 (2013)Google Scholar
  5. 5.
    T. Asselmeyer-Maluga, J. Król, Int. J. Geom. Met. Mod. Phys. 10 Article Number: 1350055 (2013)Google Scholar
  6. 6.
    P. Bankstone, J. Pure Appl. Algebra 97, 221 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    C.H. Brans, Class. Quant. Gravity 11, 1785 (1994)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    C.H. Brans, J. Math. Phys. 35, 5494 (1995)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    C.H. Brans, T. Asselmeyer-Maluga, Exotic Smoothness and Physics: Differential Topology and Spacetime Models (World Scientific, Singapore, 2007)zbMATHGoogle Scholar
  10. 10.
    A. Connes, J. Lott, Nucl. Phys. B Proc. Suppl. 18B 29 (1990)Google Scholar
  11. 11.
    A. Connes, Noncommutative Geometry (Academic Press, London, 1994)zbMATHGoogle Scholar
  12. 12.
    S. DeMichelis, M. Freedman, J. Differ. Geom. 35, 219 (1992)MathSciNetGoogle Scholar
  13. 13.
    S.K. Donaldson, J. Differ. Geom. 18, 279 (1983)MathSciNetGoogle Scholar
  14. 14.
    M. Freedman, J. Differ. Geom. 17, 357 (1982)Google Scholar
  15. 15.
    I.M. Gelfand, A.N. Kolgomorov, Dokl. Acad. Nauk SSSR 22, 11 (1939)Google Scholar
  16. 16.
    L. Gillman, M. Jerison, Rings of Continuous Functions (Springer, Berlin, 1986)zbMATHGoogle Scholar
  17. 17.
    R.E. Gompf, J. Differ. Geom. 18, 317 (1983)MathSciNetGoogle Scholar
  18. 18.
    R.E. Gompf, J. Differ. Geom. 37, 199 (1993)MathSciNetGoogle Scholar
  19. 19.
    E. Hewitt, Trans. Am. Math. Soc. 64, 45 (1948)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Król (2008). arXiv:0804.4217v2
  21. 21.
    R.S. Lubarsky (2015). arXiv:1510.00988
  22. 22.
    S. Mrówka, Acta Math. Acad. Sci. Hung. 21, 239 (1970)CrossRefGoogle Scholar
  23. 23.
    E.E. Rosinger, A. Khrennikov, AIP Conf. Proc. 03/2011 1327(1), 520–526. doi: 10.1063/1.3567483
  24. 24.
    J. Sładkowski, Acta Phys. Pol. B 27, 1649 (1996)Google Scholar
  25. 25.
    J. Sładkowski, Int. J. Mod. Phys. D 9, 311 (2001)Google Scholar
  26. 26.
    L. Sokołowski, Acta Phys. Pol. B (2015). In pressGoogle Scholar
  27. 27.
    E.S. Thomas, Trans. Am. Math. Soc. 126, 244 (1967)CrossRefGoogle Scholar
  28. 28.
    E.M. Vechtomov, Itogi Nauki i Tekhniki, ser Algebra, Topologia, Geometria (in Russian), vol. 28, p. 3 (1990)Google Scholar
  29. 29.
    E.M. Vechtomov, Itogi Nauki i Tekhniki, ser. Algebra, Topologia, Geometria (in Russian), vol. 29, p. 119 (1990)Google Scholar
  30. 30.
    J.V. Whittaker, Ann. Math. 62, 74 (1963)MathSciNetCrossRefGoogle Scholar
  31. 31.
    O. Yaremchuk, (1999). arXiv: quant-ph/9902060

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Physics, University of SilesiaKatowicePoland

Personalised recommendations