Advertisement

A Glimpse into the Problems of the Fourth Dimension

  • Valentin Poénaru
Chapter

Abstract

The message of this short survey is that four-dimensional topology is very special indeed. Also, four dimensions is the place where, today, as far as topology of manifolds is concerned, more than anywhere else, there are still big questions waiting to be solved.

References

  1. 1.
    M. Artin and B. Mazur, On periodic points, Ann of Math. 2 (1965), pp. 82–99.CrossRefGoogle Scholar
  2. 2.
    M.F. Atyiah, Geometry of Yang-Mills Fields, Lezioni Fermiane, Pisa (1979).Google Scholar
  3. 3.
    L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, Geometrization of 3-manifolds, EMS Tracts in Math. 13 (2010).Google Scholar
  4. 4.
    S.K. Donaldon, An application of gauge theory to four-dimensional topology, J. Diff. Geom. 18 (1983), pp. 278–315.MathSciNetGoogle Scholar
  5. 5.
    D.S. Freed and K.K. Uhlenbeck, Instantons and four-manifolds, Springer (1984).Google Scholar
  6. 6.
    M. Freedman, The Topology of four-dimensional manifolds, J. Diff. Geom. 17 (1983), pp. 279–315.MathSciNetGoogle Scholar
  7. 7.
    M.H. Freedman and F. Luo, Selected applications of geometry to low-dimensional topology, Marker Lectures, AMS (1990).Google Scholar
  8. 8.
    M.H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton mathematical Series 39, Princeton N.J. (1990).Google Scholar
  9. 9.
    L. Guillou and A. Marin, À la recherche de la topologie perdue. Birkhäuser 1986.Google Scholar
  10. 10.
    M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. of Math. 77 (1963), pp. 504–537.MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Mazur, On embeddings of spheres, BAMS 65 (1959), pp. 59–65.MathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. 73 (1961), pp. 221–228.MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.E. Moise, Affine structures in 3-manifolds V, the triangulation theorem and Hauptvermutung, Ann. of Math. 55 (1952), pp. 96–114.MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Morgan, The Seiberg-Witten Equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, Princeton (1996).Google Scholar
  15. 15.
    J. Morgan and G. Tian, Ricci flow and the Poincaré Conjecture, Clay Math. Monographs 3, AMS (2007).Google Scholar
  16. 16.
    V. Poénaru, Les décompositions de l’hypercube en produit topologique, Bull. Soc. Math. de France 88 (1960), pp. 113–129.MathSciNetCrossRefGoogle Scholar
  17. 17.
    V. Poénaru, Processus infinis et conjecture de Poincaré en dimension trois, IV: Le théorème de non-sauvagerie lisse (The smooth tameness theorem) Part A, Prépublications Orsay 93–83 (1993), Part B, Prépublications Orsay 95–33 (1995).Google Scholar
  18. 18.
    V. Poénaru, Geometric Simple Connectivity and Low-Dimentional Topology, Proceedings Steklov Inst. of Math. 247 (2004), pp. 195–208.zbMATHGoogle Scholar
  19. 19.
    V. Poénaru, What is … an infinite swindle, Notices AMS, Vol. 54, no 5, May 2007, pp. 619–622.Google Scholar
  20. 20.
    V. Poénaru, Geometric Simple Connectivity and finitely presented groups, arXiv:1404.4283 (math G.T.) (2014).Google Scholar
  21. 21.
    V. Poénaru and C. Tanasi, Introduzione alla geometria e alla topologia dei Campi di Yang-Mills, Sopplemento dei Rendicondiconti del Circolo Matematico di Palermo, S.2, N.13 (1986), pp. 1–55.Google Scholar
  22. 22.
    V. Poénaru and C. Tanasi, Representations of the Whitehead Manifold Wh3 and Julia Sets, Ann. Toulouse, Vol. IV, no 3 (1995), pp. 665–694.Google Scholar
  23. 23.
    S. Smale, On the Struture of Manifolds, Amer. J. of Math. 84 (1962), pp. 387–399.MathSciNetCrossRefGoogle Scholar
  24. 24.
    S. Smale, The Story of the Higher Dimensional Poincaré Conjecture (What actually happened on the beaches of Rio). Mathematical Intelligencer, Vol. 12, no 2 (1990).MathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Stallings, The piece-wise linear structure of Euclidean spaces, Proc. Cambr. Math. Soc. 58 (1962), pp. 481–488.MathSciNetCrossRefGoogle Scholar
  26. 26.
    W. Thurston, The geometry and topology of 3-manifolds, Lecture Notes, Princeton University (1980).Google Scholar
  27. 27.
    C.T.C. Wall, Classifications of handlebodies, Topology 2 (1963), pp. 263–272.MathSciNetCrossRefGoogle Scholar
  28. 28.
    E.C. Zeeman, The Poincaré Conjecture for n ≥ 5, in Topology of 3-manifolds and related topics, M.K. Ford ed., Prentice-Hall (1962).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Valentin Poénaru
    • 1
  1. 1.Professor Emeritus at the Université Paris Sud-OrsayOrsayFrance

Personalised recommendations