Smoothability of proper foliations

Cantwell, John; Conlon, Lawrence

Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 219-244.

Il a été prouvé que toutes les variétés feuilletées compactes de classe $C^2$, de codimension 1, dont toutes les feuilles sont propres, sont de classe $C^{\infty }$. Plus précisément, une telle variété feuilletée est homéomorphe à une variété de classe $C^{\infty }$. En d’autres termes, le résultat n’est pas vrai pour un feuilletage à feuilles non-propres. Dans ce cas précis, il y a une différence du point de vue topologique entre les classes $C^r$ et $C^{r+1}$, pour tout entier naturel $r$.

Compact, $C^2$-foliated manifolds of codimension one, having all leaves proper, are shown to be $C^{\infty }$-smoothable. More precisely, such a foliated manifold is homeomorphic to one of class $C^{\infty }$. The corresponding statement is false for foliations with nonproper leaves. In that case, there are topological distinctions between smoothness of class $C^r$ and of class $C^{r+1}$ for every nonnegative integer $r$.

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