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described in unpublished work of Thurston.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.
Keywords
normal surfaces, 3–manifolds, ideal triangulations
Mathematical Subject Classification
Primary: 57M25
Secondary: 57N10
Publication
Received: 8 January 2004
Revised: 29 March 2004
Accepted: 16 March 2004
Published: 20 September 2004