#### Geometry & Topology, Vol. 6 (2002)
Paper no. 6, pages 153--194.

## Laminar Branched Surfaces in 3-manifolds

### Tao Li

**Abstract**.
We define a laminar branched surface to be a branched surface
satisfying the following conditions: (1) Its horizontal boundary is
incompressible; (2) there is no monogon; (3) there is no Reeb
component; (4) there is no sink disk (after eliminating trivial
bubbles in the branched surface). The first three conditions are
standard in the theory of branched surfaces, and a sink disk is a disk
branch of the branched surface with all branch directions of its
boundary arcs pointing inwards. We will show in this paper that every
laminar branched surface carries an essential lamination, and any
essential lamination that is not a lamination by planes is carried by
a laminar branched surface. This implies that a 3-manifold contains an
essential lamination if and only if it contains a laminar branched
surface.
**Keywords**.
3-manifold, branched surface, lamination

**AMS subject classification**.
Primary: 57M50.
Secondary: 57M25, 57N10.

**DOI:** 10.2140/gt.2002.6.153

**E-print:** `arXiv:math.GT/0204012`

Submitted to GT on 16 February 2001.
(Revised 8 July 2001.)
Paper accepted 18 March 2002.
Paper published 30 March 2002.

Notes on file formats
Tao Li

Department of Mathematics, 401 Math Sciences

Oklahoma State University, Stillwater, OK 74078, USA

Email: tli@math.okstate.edu

URL: http://www.math.okstate.edu/~tli

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