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René Thom and an Anticipated h-Principle

In: Geometry in History

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  • François Laudenbach

    (Université de Nantes, Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS)

Abstract

The first part of this chapter intends to present the role played by Thom in diffusing Smale’s ideas about immersion theory, at a time (1957) where they sound counterintuitive: it isclearlyimpossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale’s idea in what is now known as the h-principle. Here, the h stands for homotopy. Shortly after the astonishing discovery by Smale, Thom gave a lecture in Lille (1959) announcing a theorem which would deserve to be named a homologicalh-principle. The aim of our second part is to comment about this theorem which, at that time, was completely ignored by the topologists in Paris, but not in Leningrad. We explain Thom’s statement and comment about it. The first idea is combinatorial. A beautiful subdivision of the standard simplex emerges from Thom’s article. We connect it with the jiggling technique introduced by W. Thurston in his seminal work on foliations.

Suggested Citation

  • François Laudenbach, 2019. "René Thom and an Anticipated h-Principle," Springer Books, in: S. G. Dani & Athanase Papadopoulos (ed.), Geometry in History, chapter 0, pages 469-491, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-13609-3_12
    DOI: 10.1007/978-3-030-13609-3_12
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