C1,0 foliation theory

Kazez, William; Roberts, Rachel

doi:10.2140/agt.2019.19.2763

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$C^{1,0}$ foliation theory

William H Kazez and Rachel Roberts

Algebraic & Geometric Topology 19 (2019) 2763–2794

DOI: 10.2140/agt.2019.19.2763

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.

Keywords

codimension-one foliation, flow, dimension-one foliation, measured foliation, holonomy, flow box decomposition, Denjoy blowup

Mathematical Subject Classification 2010

Primary: 57M50

References

Publication

Received: 10 December 2016

Revised: 22 January 2019

Accepted: 5 February 2019

Published: 20 October 2019

Authors

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

Volume 19, issue 6 (2019)