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

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

Chapter Chapter 9 in Essays on Topology, 2025, pp 135-160 from Springer

Abstract: 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.

Keywords: Diffeomorphisms of the 3-sphere; Closed 1-forms; Dehn twist; Dehn modification; 57R19 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/978-3-031-81414-3_9

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