The simplest branched surfaces for a foliation

Sandra Shields

Vol. 225 (2006), No. 1, 177–198

DOI: 10.2140/pjm.2006.225.177

Abstract

Given a foliation F of a closed 3-manifold and a Smale flow ϕ transverse to F, we associate a “simplest” branched surface with the pair (F,ϕ), which is unique up to two combinatorial moves. We show that all branched surfaces constructed from F and ϕ can be obtained from the simplest model by applying a finite sequence of these moves chosen so that each intermediate branched surface also carries F. This is used to partition foliations transverse to the same flow into countably many equivalence classes.

Keywords

branched surface, foliation, transverse flow

Mathematical Subject Classification 2000

Primary: 57R30, 57M50

Secondary: 57M10, 57M20, 57N10

Milestones

Received: 4 May 2004

Revised: 5 November 2005

Accepted: 17 March 2006

Published: 1 May 2006