ℤ2–Thurston norm and complexity of 3–manifolds, II

Abstract

In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry SL 2 (R). IntroductionGiven a closed, irreducible 3-manifold, its complexity is the minimum number of tetrahedra in a (pseudo-simplicial) triangulation of the manifold. This number agrees with the complexity defined by Matveev [12] unless the manifold is S 3 , RP 3 or L(3, 1). We denote the complexity of M by c(M). The only known infinite families of closed 3-manifolds for which an exact value of the complexity is known are spherical space forms [7,8,9].Asymptotic bounds for an infinite family of double branched coverings of certain alternating closed 3-braids are given by Ni and Wu [14] using the results of [9]. New lower bounds on the complexity of closed 3-manifolds were recently given with different methods by Cha [2, 3].In this paper, we continue to be interested in determining the exact complexity and minimal triangulations of infinite classes of closed 3-manifolds. In order to describe our results, we need the following definition.Let M be a closed, orientable, irreducible, connected 3-manifold, and let S be an embedded closed surface dual to a given ϕ ∈ H 1 (M; Z 2 ). An analogue of Thurston's norm [16] is defined in [9] as follows. If S is connected, let χ − (S) = max{0, −χ(S)}, and otherwise letwhere the sum is taken over all connected components of S. Note that S i is not necessarily orientable. Define:|| ϕ || = min{χ − (S) | S dual to ϕ}.The surface S dual to ϕ ∈ H 1 (M; Z 2 ) is said to be Z 2 -taut if no component of S is a sphere and χ(S) = −|| ϕ ||. As in [16], one observes that (after possibly deleting compressible tori) every component of a Z 2 -taut surface is non-separating and geometrically incompressible. This follows from the fact that the Klein bottle does not embed in RP 3 .

On minimal ideal triangulations of cusped hyperbolic 3‐manifolds

“…Minimal triangulations, that is, topological ideal triangulations using the least number of ideal 3-simplices, are used in census enumeration, and as a platform to study the topology of the manifold using normal surface theory. In this paper, we establish basic facts about minimal ideal triangulations of cusped hyperbolic 3-manifolds in analogy with our previous work for closed 3-manifolds [20][21][22]. Throughout this paper, by a cusped hyperbolic 3-manifold, we mean an orientable noncompact 3-manifold M that admits a complete hyperbolic structure of finite volume.…”

“…The essence of the proof is a counting argument taking into account even edges, compression discs for the canonical normal representatives and types of tetrahedra. This is modelled on the blueprint for closed 3-manifolds [20][21][22]28]. Moreover, in the case of equality the triangulation is minimal, each canonical normal representative of a non-zero element in H H 2 (M, Z 2 ) is taut and meets each tetrahedron in a quadrilateral disc, and the number of tetrahedra in the triangulation is even.…”

“…The argument for closed 3‐manifolds in hinged on an understanding of the intersections of maximal layered solid tori. We now carry such an analysis out in the case of cusped hyperbolic 3‐manifolds.…”

“…In § , we describe normal surfaces representing ‐homology classes in ideal triangulations. We use this in § to prove our main result Theorem , which is analogous to the main result of . Before we can state it, we need to introduce some extra definitions.…”

“…We wish to emphasise that we do not work with . An analogue of Thurston's norm is defined in as follows. If is connected, let , and otherwise let where the sum is taken over all connected components of .…”