Volume 6, issue 1 (2002)

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Laminar branched surfaces in $3$–manifolds

Tao Li

Geometry & Topology 6 (2002) 153–194
 arXiv: math.GT/0204012
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
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57M25, 57N10