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.

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Tao Li
Department of Mathematics, 401 Math Sciences
Oklahoma State University, Stillwater, OK 74078, USA
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