$C^{1,0}$ foliation theory

Abstract

Transverse $1$–dimensional foliations play an important role in the study of codimension-one foliations. In Geom. Topol. Monogr. 19 (2015) 21–72, the authors introduced the notion of flow box decomposition of a $3$–manifold $M$. This is a combinatorial decomposition of $M$ that reflects both the structure of a given codimension-one foliation and that of a given transverse dimension-one foliation, and that is amenable to inductive strategies.

In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically $3$–dimensional techniques, and should generalize to prove corresponding results for codimension-one foliations in $n$–dimensional manifolds.