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Ideal triangulations of 3–manifolds I: spun normal surface theory. (English) Zbl 1085.57016

Gordon, Cameron (ed.) et al., Proceedings of the Casson Fest. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences, Fayetteville, AR, USA, April 10–12, 2003 and the conference on the topology of manifolds of dimensions 3 and 4, Austin, TX, USA, May 19–21, 2003. Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 235-265 (2004).
This is the first of a series of five papers in which the authors investigate ideal triangulations of the interiors of compact \(3\)-manifolds with boundary consisting of tori and Klein bottles. This paper concerns the normal surface theory for such triangulations. Normal surfaces have been a very useful device in certain aspects of \(3\)-manifold theory, especially in the study of algorithmic solutions of fundamental problems. In the traditional theory, originated by W. Haken, one fixes a triangulation of a \(3\)-manifold and defines a normal surface to be a map of a closed surface into the \(3\)-manifold which can be pieced together from imbeddings of copies of quadrilaterals that are links of edges and triangles that are links of vertices of the tetrahedra of the triangulation. Each tetrahedron contains three kinds of quadrilaterals, corresponding to pairs of opposite edges, and four kinds of triangles, corresponding to the vertices. Any imbedded incompressible surface is isotopic to an imbedded normal surface, but in general a normal surface need not be imbedded or even immersed, as there may be branching where it meets an edge of the triangulation. For an ideal triangulation of the interior of a compact \(3\)-manifold with boundary, a properly imbedded nonclosed surface cannot be normal. Using an idea which they attribute to W. Thurston, the authors enlarge the class of normal surfaces to include spun normal surfaces, which are noncompact surfaces in the interior of the \(3\)-manifold. Near an end of the interior, a spun normal surface looks like a collection of half-open annuli that spiral around the missing boundary component. A spun normal surface has a well-defined rational boundary slope in each boundary component that it spins about. In each ideal tetrahedron of the triangulation, a spun normal surface may have infinitely many triangles, but only finitely many quadrilaterals. For \(3\)-manifolds with non-toroidal boundary components, there is a version of spun normal surfaces which meet a neighborhood of a boundary component in immersed annuli. These are used in a later paper of the series. The main task of the present paper is a calculation of the dimension of the space \(\mathcal{W}\) of spun and ordinary normal surfaces which are not parallel to a boundary component. This space allows formal spun normal surfaces, which may have real and even negative multiples of triangular and quadrilateral disk types. Also, the triangulations may be pseudo-triangulations, in which a ideal tetrahedron may have some of its own faces identified. The main result is that \(\mathcal{W}\) has real dimension \(2k+c\), where \(k\) the number of edges and \(c\) is the number of boundary components. In fact, the boundary slope homomorphism \(\partial\colon \mathcal{W}\to \mathbb{R}^{2c}\) is surjective, where \(\mathbb{R}^{2c}\) is the direct sum of the real homology groups \(H_1(T_i;\mathbb{R})\), where the \(T_i\) are the torus boundary components and the orientable double covers of the Klein bottle boundary components. Consequently, over the integers the image of the boundary slope homomorphism has finite index in the direct sum of the \(H_1(T_i;\mathbb{Z})\). A key ingredient is a careful understanding of the linear functionals determined by adding up the angles of triangles and quadrilaterals where they meet a fixed edge. In the final section, the authors give a complete analysis of the cases of the figure-\(8\) knot complement and the Gieseking manifold. In particular, they find that the integral boundary slope homomorphisms are not onto.
For the entire collection see [Zbl 1066.57002].

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q15 Triangulating manifolds

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