Ideal triangulations of 3--manifolds I: spun normal surface theory

Abstract

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map from spun normal surface theory to the homology classes of boundary loops of the 3-manifold and prove the boundary map has image of finite index. Spun normal surfaces give a natural way of representing properly embedded and immersed essential surfaces in a 3-manifold with tori and Klein bottle boundary [E Kang, `Normal surfaces in knot complements' (PhD thesis) and `Normal surfaces in non-compact 3-manifolds', preprint]. It has been conjectured that every slope in a simple knot complement can be represented by an immersed essential surface [M Baker, Ann. Inst. Fourier (Grenoble) 46 (1996) 1443-1449 and (with D Cooper) Top. Appl. 102 (2000) 239-252]. We finish by studying the boundary map for the figure-8 knot space and for the Gieseking manifold, using their natural simplest ideal triangulations. Some potential applications of the boundary map to the study of boundary slopes of immersed essential surfaces are discussed.