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$C^{1,0}$ foliation theory

William H Kazez and Rachel Roberts

Algebraic & Geometric Topology 19 (2019) 2763–2794

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 C2. 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.

codimension-one foliation, flow, dimension-one foliation, measured foliation, holonomy, flow box decomposition, Denjoy blowup
Mathematical Subject Classification 2010
Primary: 57M50
Received: 10 December 2016
Revised: 22 January 2019
Accepted: 5 February 2019
Published: 20 October 2019
William H Kazez
Department of Mathematics
University of Georgia
Athens, GA
United States
Rachel Roberts
Department of Mathematics
Washington University
St Louis, MO
United States