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About the Diffeomorphisms of the 3-Sphere and a Famous Theorem of Cerf ( Γ 4 = 0 $$\Gamma _4=0$$ )

In: Essays on Topology

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  • François Laudenbach

    (Laboratoire de mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences et Techniques, Université de Nantes)

Abstract

At the Golden Age of Differential Topology, for every dimension, the question of exotic spheres arose. A sub-question emerged: does there exist in dimension n an exotic sphere just obtained by gluing two n-balls along their boundaries? Such spheres, up to diffeomorphism, form a group for the connected sum denoted by Γ n $$\Gamma _n$$ . The first discovery of this type of exotic spheres goes back to J. Milnor (1956): he proved that Γ 7 $$\Gamma _7$$ is not trivial. By proving Γ 4 = 0 $$\Gamma _4=0$$ , J. Cerf (1968) stated that the gluing of two four-dimensional balls always gives rise to the standard S 4 $$S^4$$ . Actually, Cerf proved that every diffeomorphism of S 3 $$S^3$$ is isotopic to a linear diffeomorphism. In this chapter we present a foliated proof of Cerf’s theorem.

Suggested Citation

  • François Laudenbach, 2025. "About the Diffeomorphisms of the 3-Sphere and a Famous Theorem of Cerf ( Γ 4 = 0 $$\Gamma _4=0$$ )," Springer Books, in: Louis Funar & Athanase Papadopoulos (ed.), Essays on Topology, chapter 0, pages 135-160, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-81414-3_9
    DOI: 10.1007/978-3-031-81414-3_9
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