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Spun normal surfaces in 3-manifolds. III: Boundary slopes. (English) Zbl 1497.57022

A spun normal surface in an ideally triangulated 3-manifold is a generalization of a normal surface. It contains infinitely many normal triangular disks. Let \(M\) be the interior of a compact 3-manifold \(\overline{M}\) with boundary \(\partial M\) consisting of tori and Klein bottles. Fix an ideal triangulation \(\mathfrak{T}\) of \(M\). Then a spun normal surface \(\Sigma\) in \(M\) contains infinitely many normal tori or Klein bottles parallel to a boundary component, or it has an end which spins infinitely around a boundary component. In the latter case, \(\Sigma\) represents a bounded surface in \(\overline{M}\). Then it determines the boundary slope in \(\partial M\).
Let \(\Sigma_{1}\) and \(\Sigma_{2}\) be spun normal surfaces which intersect transversely. Suppose that they do not contain normal quadrilateral disks of different types in a tetrahedron. Then the Haken sum \(\Sigma = \Sigma_{1} + \Sigma_{2}\) is defined by a regular exchange along the intersection.
The system of \(Q\)-matching equations for \((M, \mathfrak{T})\) describes constraints required to form a normal or spun normal surface from a collection of quadrilateral disks. Let \(\mathcal{W}\) denote the vector space of formal normal or spun normal solutions, which may contain normal quadrilateral disks of different types in a tetrahedron. The projective solution space \(\overline{\mathcal{W}_{s}}\) is a compact convex linear cell obtained by projectivizing a subspace of \(\mathcal{W}\) at boundary slopes \(s\).
The main results in this paper are analogous to results for normal surfaces by W. Jaco and U. Oertel [Topology 23, 195–209 (1984; Zbl 0545.57003)]. Let \(M\) be an irreducible, \(P^{2}\)-irreducible, atoroidal and anannular 3-manifold. (Equivalently, \(M\) admits a hyperbolic metric of finite volume.) Fix an ideal 1-efficient triangulation of \(M\). Let \(F\) be a proper essential surface which is not a fiber of a bundle structure of \(M\). Let \(\Sigma\) be a least-weight spun normal representative of \(F\). Suppose that \(\Sigma = \Sigma_{1} + \Sigma_{2}\) is a Haken sum, where \(\Sigma_{1}\) and \(\Sigma_{2}\) have boundary slopes within that of \(\Sigma\). Then \(\Sigma_{1}\) and \(\Sigma_{2}\) are essential surfaces. The notion of least-weight for a spun normal surface requires that the represented surface is not a fiber. Consequently, there is a set of essential spun normal surfaces at vertices of \(\overline{\mathcal{W}_{s}}\) which together have boundary slopes \(s_{i}\) for \(1 \leq i \leq k\) so that \(s = \bigcup_{1 \leq i \leq k} s_{i}\). This is a partial answer to a question by N. M. Dunfield and S. Garoufalidis [Trans. Am. Math. Soc. 364, No. 11, 6109–6137 (2012; Zbl 1281.57012)].

MSC:

57K30 General topology of 3-manifolds
57M99 General low-dimensional topology
57M10 Covering spaces and low-dimensional topology

Software:

Regina
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References:

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