h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness

Kartoue Demdah, Mady

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1573-1597.

Le théorème de h-cobordisme est bien connu en topologie différentielle et PL. Une généralisation pour les h-cobordismes possiblement non simplement connexe est appelée théorème de s-cobordisme. Dans ce papier, nous démontrons les versions semi-algébrique et Nash de ces théorèmes. C’est-à-dire, avec des données semi-algébriques ou Nash, nous obtenons un homéomophisme semi-algébrique (respectivement un difféomorphisme Nash). Les principaux outils intervenant sont la triangulation semi-algébrique et les approximations Nash.

Un aspect de la nature algébrique des objets semi-algébriques et Nash est qu’on peut mesurer leurs complexités. Nous montrons les théorèmes de h et s-cobordisme avec borne uniforme sur la complexité de l’homéomorphisme semi-algébrique (difféomorphisme Nash) obtenu, en fonction de complexité des données du cobordisme. La borne uniforme pour le h-cobordisme semi-algébrique réelle ne peut être effective. Ce qui donne un autre exemple de non effectivité en géométrie algébrique réelle. Pour finir, nous déduisons la validité de ces théorèmes version semi-algébrique et Nash sur tout corps réel clos.

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nash cobordism data, we get a semialgebraic homeomorphism (respectively a Nash diffeomorphism). The main tools used are semialgebraic triangulation and Nash approximation.

One aspect of the algebraic nature of semialgebraic or Nash objects is that one can measure their complexities. We show h and s-cobordism theorems with a uniform bound on the complexity of the semialgebraic homeomorphism (or Nash diffeomorphism) obtained in terms of the complexity of the cobordism data. The uniform bound of semialgebraic h-cobordism cannot be recursive, which gives another example of non effectiveness in real algebraic geometry. Finally we deduce the validity of the semialgebraic and Nash versions of these theorems over any real closed field.

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