  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surfaces have been described in unpublished work of Thurston.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.
