Matveev complexity of some lens spaces

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che non mi dice mai di non andare, ma mi d`a sempre la pi`u bella ragione per tornare indietro.

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Introduction

This dissertation is devoted to the study of the complexity of 3-manifolds; we shall define a function on the set of 3-manifolds onto the natural numbers which is suitable to encode, in a sense, how complicated a manifold is. The na¨ıf definition of the complexity of a 3-manifold M is the minimal number c(M ) of tetrahedra in a triangulation of M , where the notion of triangulation is meant in a loose sense, namely, self and multiple adjacencies are allowed.

The actual definition of c(M ) was given by Matveev in [Matb] as the minimum number of vertices of an almost special spine. Matveev also estab-lished the following result, which explains the relation of the actual definition with the na¨ıf version.

Theorem. c is additive under connected sum. For a closed, prime, orientable M , either c(M ) = 0 and M ∈ {S3, P3, L(3, 1), S2 × S1_{}, or c(M ) is the}

minimal number of tetrahedra.

As a consequence of the definition, it is clear that we can easily obtain upper bounds for the complexity of a 3-manifold — it is enough to exhibit a triangulation of the manifold. On the other hand, the problem of finding a lower bound is much more tricky.

Censuses of 3-manifolds having complexity upper bounded by a fixed integer k0 are available (see [MP01] as an example with k0 = 9), as well

as conjectural formulas regarding the complexity for certain families of 3-manifolds such as lens spaces, see [Mata].

The main purpose of this thesis is to prove several theorems stating that the value of the complexity of lens spaces conjectured by Matveev is correct for infinitely many of them.

The dissertation is organized in three chapters: in the first one, some preliminary notions will be given. More exactly, we shall define the notion of spine, a fundamental one in the context of the topology of 3-manifolds, and we will refine it introducing the notions of simple and special spine. Moreover, we will describe some operations on spines, and how spines may be used to investigate the topology of 3-manifolds. Then, we will formally define the

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complexity function advertised before, and we will show that it has some nice properties. Furthermore, we will describe some configurations of a spine that prevent it to realize the complexity. Then we will define lens spaces, the class of 3-manifold we will be interested in through the entire dissertation; we will describe spines for them, and we will state a conjectural formula by Matveev concerning their complexity. More exactly, we will describe a machinery to build a spine for every lens space, which is the candidate to realize the minimum of the complexity function. Finally, we will give an introduction to Haken’s theory of normal surfaces, that will be largely used in the subsequent chapters.

The second chapter is devoted to the study of an alternative approach of the theory of complexity, that uses the language of triangulations instead of spines. We shall define and describe a special family of triangulations of lens spaces, called layered triangulations, see [JR06], that can be built imposing suitable restrictions on the combinatories of the gluing of tetrahedra. It turns out that this family of triangulations is dual to the family of spines built via Matveev’s machinery. We will see that the restrictions on them make layered triangulations easily encodable using a proper graph, and we will prove that for any lens space there exists a unique layered triangulation which is minimal among all the layered triangulations of the same space.

In the last chapter we will describe a series of results of Jaco, Rubinstein and Tillmann (2009), see [JRT09], leading to the exact computation of the complexity of an infinite family of lens spaces. In their work the three authors show that the minimal layered triangulation of a lens space belonging to some specific families is minimal over all possible triangulations, i.e., it realizes the complexity. The proofs are quite technical, and use the approach to complexity via triangulations. However, whenever it is possible and it seems convenient, we reinterpret their results on triangulations in the dual language of spines, which allows in our opinion a better topological insight.

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Chapter 1 Preliminary notions

In this first chapter we shall introduce some basic notions that will be nec-essary in the sequel. More precisely, we shall define the important notion of spine of a 3-manifold, an object which is useful for the description of a 3-manifold and for the investigation of its properties, and we will use spines to define the complexity of a 3-manifold. Further on, we shall define lens spaces, and provide the reader with a conjectural formula for their complex-ity due to Matveev [Matb]; this conjectural formula will be the main theme of the entire dissertation. Eventually, we will introduce some basics of Haken’s theory of normal surfaces.

1.1 A primer on PL-Topology

Through the entire dissertation we will work in the P L-category, and we will follow the P L-approach to the topology of 3-manifolds, see [RS12]. In this section we shall define the central notion of spine of a 3-manifold, a sub-polyhedron onto which the 3-manifold collapses. In particular, we will refine the notion of spine, introducing special spines, which give a presentation of 3-manifolds.

1.1.1 Collapse

We now introduce the concept of collapse of a simplicial complex.

Definition 1.1.1. Let K be a simplicial complex, and let σn and δn−1 be two open simplices in K such that σ is not a proper face of any simplex in K, and δ is a free face of σ, i.e., δ is a face of σ and is not a proper face of any simplex in K other than σ. We call the transition from K to K \ (σ ∪ δ) an elementary simplicial collapse, see Figure 1.1.

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Figure 1.1: Elementary simplicial collapse.

Figure 1.2: Examples of collapses.

Definition 1.1.2. A polyhedron P collapses to a sub-polyhedron Q if for some triangulation (K, L) of the pair (P, Q) the complex K collapses onto L by a sequence of elementary simplicial collapses; in this case we will write P & Q.

1.1.2 Spines

Definition 1.1.3. Let M be a compact 3-manifold, with ∂M 6= ∅. A sub-polyhedron P ⊂ M is called a spine of M if M & P , i.e., M collapses onto P . If M is a closed, connected 3-manifold, by a spine of M we mean a spine of the 3-manifold with boundary M \ B3_{, where B}3 _{is an open 3-ball in M . If}

M is a disconnected 3-manifold, we call a spine of M the union of one spine for each connected component of M .

Proposition 1.1.4. Any compact 3-manifold has a spine of dimension less then or equal to 2.

Proof. Without loss of generality, we can assume that M is connected and with boundary. Let us realize M as a simplicial complex, i.e., let us tri-angulate it. Let K be a sub-complex obtained via a maximal sequence of collapses. We claim that K has dimension less or equal to two. Arguing by contradiction, if K contains at least one 3-simplex, then there exists a 3-simplex with a free facet and a further collapse can be performed, against our assumptions on K.

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Figure 1.3: The annulus and the M¨obius strip have homeomorphic spines, even if they are not homomorphic 2-manifolds.

The following theorem gives some equivalent definitions of the notion of spine; first we will recall the next definition:

Definition 1.1.5. Let f : X −→ Y be a map between two topological spaces. The mapping cylinder Cf is defined as Y ∪ (X × [0, 1])/ ∼, where the

equivalence relation is generated by identifications (x, 1) = f (x) for every x ∈ X.

Theorem 1.1.6 ([Mata]). Let M be a compact manifold with boundary. The following conditions on a compact sub-polyhedron P ⊂ IntM are equivalent to each other:

1. P is a spine of M ;

2. M is homeomorphic to a regular neighbourhood of P in M ;

3. M is homeomorphic to the mapping cylinder of a map f : ∂M −→ P ; 4. the manifold M \ P is homeomorphic to ∂M × [0, 1).

The definition easily implies that if Q is a spine of P then P deformation retracts onto Q, so Q is homotopy equivalent to P . However, Q may not determine the homeomorphism type of P , as shown in Figure 1.3 (the annulus and the M¨obius strip both have a circle as a spine). For 3-manifolds, this drawback can be circumvented by imposing topological constraints on the spine. We will give the precise definitions shorty later. Before, we have to define other fundamental notions.

Definition 1.1.7. A compact polyedron P is called simple if the link of each point in P is homeomorphic to one of the following:

1. A circle; in this case the point is called non-singular ;

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Figure 1.4: Typical neighbourhoods of a point in a simple polyhedron. From left to right: a non-singular point, a triple point, and a true vertex.

Figure 1.5: Three equivalent ways to represent the butterfly.

3. A circle with three radii; in this last case the point is called a true vertex, or more briefly a vertex.

In Figure 1.4 we show regular neighbourhoods of points in a simple poly-hedron. The regular neighbourhood of a vertex is called butterfly, and alter-native pictures of it are shown in Figure 1.5. The right most model is the union of the links of the vertices in the first barycentric subdivision of the tetrahedron.

Definition 1.1.8. We will call triple points and true vertices singular points. The set of all singular points of a simple polyhedron P is called its singular graph, and denoted by S(P ).

It follows from the definition that each simple polyhedron has a natural stratification, where strata of dimension 2 are made of non-singular points, strata of dimension 1 are made of triple points, and each stratum of dimension 0 is a vertex. Hence, the strata of dimension 2 are open surfaces, those of dimension 1 are arcs or closed curves, and those of dimension 0 are isolated points. A simple polyhedron may not be cellular. However, we can refine the notion of simple polyhedron requiring it to be cellular: this is formalized in the next definition.

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Figure 1.6: Subtituiting each piece of the triangulation with a handle.

Figure 1.7: Going from a triangulation to a handle decomposition.

1. Each 1-dimensional stratum of P is an open 1-cell. 2. Each 2-dimensional stratum of P is an open 2-cell.

Definition 1.1.10. A spine of a 3-manifold is called simple or special if it is a simple or special polyhedron, respectively.

We shall see that special spines are the suitable tool for the study of 3-manifolds. We begin with the following:

Theorem 1.1.11. Any closed 3-manifold has a special spine.

Proof. The proof is organized as follows: first we will construct a special spine P for the manifold M with a certain number of balls removed. Then, we will prove that if M with n > 1 balls removed has a special spine, then M with n − 1 balls removed also has a special spine. Let us consider a simplicial triangulation T of M , and construct a handle decomposition associated to it. We will proceed as shown in Figure 1.6, by replacing each vertex of T with a ball Bi, each edge with a 1-handle Cj, and each triangle with a 2-handle

Dk. The result is shown in Figure 1.7. The rest of M consists of 3-handles.

We consider the following sub-polyhedron of M : P = [

i,j,k

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Figure 1.8: The Bing’s house with two rooms: this is a special spine of the 3-ball.

Figure 1.9: A sequenze of lune moves can be applied in order to make the polyhedron cellular.

Note that the union is taken all over the handles of index 0, 1 and 2, while those of index 3 are not involved in the union. The resulting polyhedron P is clearly simple, and M with a ball removed from each handle collapses on it. Furthermore, it is known by [Cas65] that for each chamber of P there exists an homeomorphism of the Bing’s house with two rooms into it, see Figure 1.8. Hence M collapses on P with a certain number of Bing’s houses added. We denote by P0 the resulting polyhedron. P0 may not be cellular, since the Bing’s room with the 1-skeleton removed is the union of three discs. However, the 2-skeleton of P0 is the union of a finite number of open discs with a finite number of closed discs removed. We can remove these holes by applying a finite sequence of lune moves, see Figure 1.9, i.e., by changing the way how the chambers of the decomposition are glued. This way, we finally obtain a special polyhedron on which M with a ball removed from each handle collapses.

Now we shall prove that if M with n > 1 balls removed has a special spine, then M with n − 1 balls removed also does; in doing this, we will define the arch construction, which will give us the solution. We accomplish our task

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A

C

B

_C

D

E

Figure 1.10: The arch construction. The arch is built as follows: we choose a point in ∂C and we act near it as described in the picture. We add to the spine a membrane E and an empty tube, so that the two distinct components A and B are fused in a unique component D.

in two steps; first, we show that as long as the number of removed balls is at least two, there exist two distinct balls separated by a component of P of dimension 2; for this, it is sufficient to observe that a general position arc connecting two distinct balls must pass transversely through at least one 2-component. Now we would like to make a hole in the spine P to fuse the two balls into a unique ball, preserving the property of the spine of being special. In order to achieve this task, we cannot simply puncture the special spine P , since the boundary of the hole would contain forbidden points with regard to the definition of special spine. In order to avoid this problem, we will use the arch construction advertised before, see Figure 1.10. We claim that this way we obtain a spine for the manifold M with the number of removed balls decreased by one; to do this, we shall prove that the new polyhedron is still cellular. The only 2-component which can possibly cause problems is that named D in Figure 1.10, that appears after the fusion of the two components A and B via the arch. The component D is indeed a 2-cell provided that the two old components A and B are distinct. However, this always happens; indeed, we started by considering two distinct balls separated by the same component C: then A must differ from B, since they separate two distinct balls. We carry on inductively, by making fusions between removed balls, until we get a special spine P0 of the manifold M with a unique ball removed. Since M is closed, we are done due to the definition of special spine for a closed manifold.

The next lemma and the subsequent theorem justify our interest in special spines.

Lemma 1.1.12 (Casler, 1965, [Cas65]). Let Pi ⊂ Mibe special sub-polyhedron

of a 3-manifold Mi for i = 1, 2. Then any homeomorphism h : P1 −→ P2

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Theorem 1.1.13. If two compact connected 3-manifolds have homeomor-phic special spines and either both are closed or both have non-empty bound-ary, then these 3-manifolds are homemorphic.

By Theorem 1.1.13, a special spine P of a 3-manifold M determines M up to homeomorphism. However, not every special polyedron P is a spine of some M . If this happens, we say that P is thickenable. An intrinsic charac-terization of the thickenable P ’s is available in [Mata].

Before dealing with the definition of the complexity, we shall recall some further definitions.

Definition 1.1.14. The connected sum M1#M2 of two compact 3-manifolds

M1 and M2 is defined as the manifold (M1\ Int B1) ∪h(M2\ Int B2), where

B1 ⊂ M1 and B2 ⊂ M2 are 3-balls, and h is a homeomorphism between

their boundaries. If the two manifolds are orientable, we require h to be an orientation-reversing diffeomorphism.

Remark. The sphere is the neutral element with respect to the connected sum, i.e., M #S3 _{= M .}

Definition 1.1.15. A connected sum M1#M2 is trivial if either M1 or M2

is a sphere.

Definition 1.1.16. Let M1 and M2 be two compact manifolds with

bound-ary, and consider two discs D1 ⊂ ∂M1 and D2 ⊂ ∂M2. The manifold M

obtained by gluing M1 and M2 together by identifying the discs along a

homeomorphism h : D1 −→ D2 is called the boundary connected sum of M1

and M2, and is denoted by M1 ⊥ M2. The resulting manifold may depend

both on the choice of the discs, and on the choice of h. However, we shall use the notation M = M1 ⊥ M2 to mean that M is one of the possible manifolds

that we can obtain via a gluing as described.

1.2 Complexity of 3-manifolds

A good notion of “complexity” for a 3-manifold should have the following properties:

• manifolds with higher complexity should be “ more complicated ” than manifolds with lower complexity;

• only finitely many manifolds should have complexity up to any preas-signed upper bound;

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Figure 1.11: Bing’s house presented as a neighbourhood of its singular graph.

• the complexity function should be additive under connected sum. The Heegard genus and the minimal number of tetrahedra in a triangulation both satisfy the first of these properties, but not the second one. For this reason, we will define complexity via spines, following Matveev [Matb].

1.2.1 Almost simple polyhedra

In the previous section we have observed that a special spine P of a manifold M uniquely determines M . Furthermore, a regular neighbourhood N (S(P )) of the singular set of P determines P since P is obtained by attaching a 2-cell to each circle in ∂(N (S(P )). Moreover, if M is orientable then N (S(P )) embeds in R3, so we can represent P simply by drawing a picture, see Figure 1.11.

The proof of the following theorem is particularly informative in that it clarifies in which sense a picture as shown in Figure 1.8 can be thought as a special spine of a manifold.

Theorem 1.2.1. ll For any integer k there exists only a finite number of special spines with k true vertices.

Proof. We shall describe a finite set of polyhedra, that a fortiori contains all special spines having k vertices. We know a special spine P is determined by N (S(P )). Denoted by v(P ) the set of vertices of P , if |v(P )| = k then S(P ) is a 4-valent graph with k vertices, so there are finitely many possibilities for it. For v ∈ v(P ), we know that N (v) is unique —the butterfly. We will call triods the neighbourhoods in the boundary of the butterfly of triple points. So to reconstruct N (S(P )) from N (v(P )) we must choose for each of the 2k edges e of S(P ) one of the 6 possible ways to match the triods shown in Figure 1.12, and the conclusion follows.

Theorem 1.2.1 may suggest to define the complexity of a 3-manifold M as the minimum number of vertices all over the special spines of M .

However, the complexity function defined via special spines is not the tool we wished to have: it is not additive under connected sum, and if we

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Figure 1.12: Gluing the two triods amounts to choosing a pairing between the two sets {1, 2, 3}.

focus only on special spines we cannot consider some very natural spines, as the point for the ball, a circle for the solid torus and the projective plane for RP3_{. All these shortcomings have the same root: a sub-polyhedron of}

a special polyhedron may not be special, even if we perform on it all the further possible collapses. For this reason, we shall define a wider class of polyhedra, properly containing special polyhedra, which is the minimal class of polyhedra closed with respect to the passage to sub-polyedra.

Definition 1.2.2. A compact polyhedron P is almost simple if the link of any of its point embeds into Γ4, the complete graph with four vertices.

Observation. It follows immediately from the definition that the property of being an almost simple polyhedron is preserved when we pass to sub-polyhedra.

Notice that Γ4 is a circle with three radii, which coincides with the

bound-ary of the butterfly. For our purposes we are usually interested in almost simple polyhedra that cannot be collapsed onto smaller sub-polyhedra. No-tice that any sub-polyhedron of Γ4 collapses onto a polyhedron Q of one of

the following types:

1. Q is either empty or a finite set of n ≥ 2 points.

2. Q is the union of a finite set (possibly empty) and a circle.

3. Q is the union of a finite set (possibly empty) and a circle with a diameter.

4. Q is the entire Γ4.

Thus, an almost simple polyhedron cannot be further collapsed onto a smaller polyhedron if and only if the link of any of its point appears in the list above. It follows from the definition that any point of an almost simple polyhedron has a neighbourhood homeomorphic to a polyhedron of dimension ≤ 1 or

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to the union of such a polyhedron with the cone over the circle, or over the circle with two or three radii.

Definition 1.2.3. A point of an almost special polyhedron is called a vertex if its link is Γ4.

1.3 The complexity function

The complexity function advertised at the beginning of the section can now be defined.

Definition 1.3.1. The complexity c(P ) of an almost simple polyhedron P is equal to the number of its vertices.

Definition 1.3.2. Let M be a compact 3-manifold; we say that M has complexity k, and we write c(M ) = k, if M has an almost simple spine with k vertices, and M has no almost simple spines with a smaller number of vertices. More briefly, c(M ) = minPc(P ), where the minimum is taken over

all the almost simple spines of M .

Let us give some simple examples. The complexity of S3, of the projective space RP3_{, of the manifold S}2_{× S}1 _{and of the lens space L}

3,1is equal to zero,

as they have almost simple spines having no vertices: respectively, the point, the projective plane RP2, the wedge of S2 with S1, and the manifold shown in Figure 1.13. It follows from the definition that it is rather easy to find an upper estimate on the complexity, as it is enough to exhibit an almost special spine of the manifold; whereas it is much more tricky to find lower estimates on the complexity, and the problem of calculating exactly the complexity of a 3-manifold is very difficult.

Mateveev showed that for a closed irreducible manifold M with positive complexity any minimal spine P of M , i.e., an almost simple spine of M realizing c(M ) that cannot be further collapsed, is special. Furthermore, the triangulation of M dual to P shown in Figure 1.14 has the minimal possible number of tetrahedra among all triangulations of M .

As it will be clear in the following, we will prefer Matveev’s theory of complexity, defined through almost simple spines, because of its hight flexi-bility. Shortly next we will state and prove some results that highlight what we mean by saying that spines are rather flexible to handle.

Proposition 1.3.3. Suppose that B is a 3-ball in a 3-manifold M . Then c(M ) = c(M \ IntB).

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Figure 1.13: A spine of the 3-manifold L(3, 1); it has no vertices. The quotient space of the circle with the identifications on its boundary shown below is equiva-lent to the union of the two objects at the bottom of the figure.

Figure 1.14: Duality between special spines and triangulations of 3-manifolds; a true vertex correspond to a tetrahedron, and vice versa.

Proof. See Figure 1.15.

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1.4 Properties of complexity

The class of almost simple polyhedra has the advantage of being closed under taking sub-polyhedra. However, an almost simple spine of a 3-manifold M does not determine M , whereas a special one does. So it would be desirable to replace an almost simple spine with a special one without increasing the number of vertices. It turns out this can be done without affecting the estimate on complexity in a specific but very relevant situation, as we will now explain.

Definition 1.4.1. A 3-manifold M is called irreducible if every 2-sphere in M bounds a 3-ball. Otherwise, M is called reducible.

It is known that if M is reducible, then either it can be decomposed into non-trivial connected sum (i.e., M is not prime), or it is one of the manifolds S2× S1 _{or S}2

e ×S1_.

Definition 1.4.2. A compact surface F in a 3-manifold M is called proper if F ∩ ∂M = ∂F.

Definition 1.4.3. A 3-manifold M is boundary-irreducible if for every proper disc D ⊂ M the curve ∂D bounds a disc in ∂M .

Definition 1.4.4. Let M be a 3-manifold with boundary. A proper annulus A ⊂ M is parallel to the boundary if there exists a annulus A0 ⊂ ∂M with ∂A0 = ∂A and A ∪ A0 _{= ∂T, see Figure 1.16.}

A

Figure 1.16: An annulus A parallel to the boundary.

Definition 1.4.5. Let M be an irreducible and boundary-irreducible man-ifold. A proper annulus A ⊂ M is called inessential if it is parallel to the boundary, or the core circle of A is contractible in M . Otherwise, A is called essential.

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We now state the important theorem which establishes suitable conditions under which an almost special spine can be converted in a special spine. Theorem 1.4.6. Suppose M is a compact, irreducible and boundary irre-ducible 3-manifold such that M 6= D3_{, S}3_{, RP}3_{, L}

3,1 and all proper annuli in

M are inessential. Then for any almost simple spine P of M there exists a special spine P0 of M having as many as or fewer vertices as P .

Idea of the proof. The proof of this result is quite sophisticated, so here we will give only a sketch of it; full details can be found in [Mata]. The idea of the proof is the following: we shall introduce a triple of numbers associated to any spine of M and depending on the topology of the corresponding spine, and we order them lexicographically. Then, we define three moves which, if performed on a spine Q of the manifold, give rise to a spine Q0 of the same manifold, whose corresponding triple is less than the previous one. We carry on the procedure until no further move can be applied. Eventually we obtain a new spine of M which has not 1-dimensional portion, and which is cellular. This way we obtain the desired special spine.

We finally state, without proving it, the result concerning additivity of complexity with respect to connected sum and boundary connected sum. Theorem 1.4.7. For any 3-manifolds M1 and M2 we have:

1. c(M1#M2) = c(M1) + c(M2);

2. c(M1 ⊥ M2) = c(M1) + c(M2).

1.5 Non-minimality criteria

We describe here two moves, that allow us to deduce that a special spine with certain properties is not minimal.

Definition 1.5.1. Let P be a special polyhedron, and c be one of its components. Since P is cellular, c can be viewed as the result of attaching a 2-dimensional disc D2_{along a certain map f : ∂D}2 _{−→ S(P ). The curve f (∂D)}

will be called the boundary curve of c. We say that f (∂D) has a counterpass if it passes along one of the edges of P twice in opposite directions; that it is short if it passes through no more than 3 vertices of P , and through each of them no more than once; that it has a triple overpass if it passes trice in the same direction along an edge of P .

Proposition 1.5.2. Suppose that P is a special spine of the closed 3-manifold M such that either:

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P

Figure 1.17: The cell representing the side wall of the cylinder is a 2-component with a counterpass.

1. M is closed and orientable, and the boundary curve of one of the 2-components of P has a counterpass.

2. P has a 2-component with a short boundary curve, 3. P has a triple overpass.

Then M possesses an almost simple spine with fewer vertices than P . Proof. Let us analyse the first case, see Figure 1.17. We shall describe a con-struction that allows us to build another spine of M having fewer vertices. Consider a disc D whose boundary ∂D is contained in the 2-cell and inter-sects S(P ). Furthermore, we can assume that the interior part of D does not intersect the polyhedron P . Now add the disc D to P ; then, there exists a ball B in M \ P which is split by D in two balls B1 and B2. The union

P ∪ D is a special spine for the twice punctured M , that is of M with the two balls B1 and B2 removed. To get a spine of M , we make a hole in one

of the two side walls V1 and V2 shown in Figure 1.17, depending on which

of them is a common face of B1 and B2. This way, we fuse the two balls,

thus obtaining another spine of the manifold M , see Figure 1.18. Moreover, if after puncturing we collapse the resulting polyhedron, the new vertex v added with the gluing of D disappears, together with at least another old vertex which is an endpoint of a singular edge starting in v. Thus we obtain a spine P0 having fewer vertices than P , and P was not minimal.

In the second case, we add a disc D, parallel to c; then D separates a ball B ⊂ M \ P in two balls B1 and B2. Now we puncture the side wall

of the cylinder determined by the two parallel copies of c, and collapse the resulting polyhedron, see Figure 1.19. After the collapse has been performed, for k = 1 at least two vertices disappears, while for k = 2 or k = 3 at least four vertices are eliminated. In any case, the resulting special polyhedron P0 has fewer vertices than P , and P was not minimal, see Figure 1.20.

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D V

V

1

2

Figure 1.18: Removing a counterpass.

Figure 1.19: A short boundary curve passing three times through a vertex. In the last case, the spine P appears as shown in Figure 1.21. Now consider the disc corresponding to the shadowed region in Figure 1.21. We can represent abstractly this portion of spine, as shown in Figure 1.22. Now by puncturing a suitable region of the hexagon in Figure 1.22 we delete two vertices (labelled v1 and v2) and add a new vertex v3. This way we obtain a

spine having fewer vertices, against minimality.

The last two moves shown in Figure 1.20 deserve a name; we will call the second one lune-move and the third one MP-move.

1.6 Lens spaces and their complexity

Here we shall recall the definition of lens space, and we will present the conjectural formula for the complexity of lens spaces proposed by Matveev. Actually, we will build a special spine realizing the conjectural complexity.

1.6.1 Lens spaces: definition and properties

We now define lens spaces, the first completely classified class of 3-manifolds, that will play a fundamental role in the following chapters.

Definition 1.6.1. Let M be a 3-manifold and T ⊂ M be a toric boundary component. A Dehn filling of M along T is the operation of gluing a solid

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Figure 1.20: Non minimality criteria: the three figures show three moves that allow us to simplify a spine which presents specific configurations. Notice that only the last move preserves the property of the spine of being special.

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v

₁

v

₃

v

₂

Figure 1.22: The boundary curve with a triple overpass and the disc. torus D × S1 _{to M via a diffeomorphism ϕ : ∂D × S}1 _{−→ T. The result}

of the operation in a new manifold. During this operation, the closed curve ∂D × {x} is glued to some simple curve γ ⊂ T, and the resulting manifold depends only on the homotopy class of γ. We will then denote it by M (γ).

Informally, we will say that the Dehn filling kills the curve γ, which is what actually happens in the fundamental group, as we will see next. First remember that, given a group G and an element g ∈ G, the normalizer of g is defined as the smallest normal subgroup of G containing g, and is denoted by N (g). Since the normalizer N (g) depends only on the conjugacy class of g, given a non-trivial, simple, closed curve γ it makes sense to consider the subgroup N ([γ]) / π1(M (γ)) without fixing a base point. The Van Kampen

Theorem easily implies the following:

Proposition 1.6.2. Let M be a 3-manifold and T ⊂ M be a boundary torus component. Denote by Mf ill

ϕ the 3-manifold obtained via a Dehn filling killing

the curve γ. Then

π1(M (γ)) ∼= π1(M )/N ([γ]).

The simplest manifold onto which one can perform a Dehn filling is the solid torus T = D2 _{× S}1_{; and a lens space can be defined as the resulting}

manifold. Fix a basis {µ, λ} for H1(∂T, Z), where µ = ∂D2 × {x} and

λ = {y} × S1_.

Definition 1.6.3. Let p and q be coprime integers; we define the lens space L(p, q) as

L(p, q) = T(qµ + pλ).

Esemples. Let us describe some simple manifolds which can be obtained as a lens space:

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• L(0, 1) = S2_{× S}1_;

• L(1, 0) = S3_;

• L(2, 1) = RP3_.

Proposition 1.6.4.

π1(L(p, q)) = Z/pZ.

Proof. It follows immediately from Proposition 1.6.2. It is also useful to state the following:

Theorem 1.6.5. Two lens spaces Lp,q and Lp0_,q0 are homemorphic if and only if

p = p0 and q ≡ ±q0 mod p.

Hence, in the following we will always suppose that p are q are coprime integers, with either p = 0 and q = 1 or p > 0 and 0 ≤ q < p.

1.6.2 The conjectural formula

In this section we shall give an estimate on complexity of lens space, which has been conjectured to be sharp by Matveev.

Continued fractions

We shall recall briefly the notion of continued fraction, which will be used in the statement of the conjectural formula for the complexity of lens spaces. Definition 1.6.6. We call continued fraction representation of a real number x an expression of type: x = a0+ 1 a1+ 1 a2+ 1 a3+ 1 a4+ . . .

where the ai’s are non-zero integers. The terms ai may possibly be infinite

in number.

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Theorem 1.6.7. For every real number x 6= 0 there exists a unique continued fraction whose value is x. Moreover, this fraction is finite if x is rational, and infinite if x is irrational.

As a consequence of the last theorem, it makes sense to give the following definition.

Definition 1.6.8. Let x be a rational number; we will call the length of its expansion in continued fraction, and denote by E(p, q), the number of terms ai that appear in its unique representation as a continued fraction.

The next example shows how to compute the expansion as a continued fraction of a rational number, and highlights the connection between the definition of continued fraction and the well-known Euclidean algorithm. Example. Consider the rational number x = 337

89; its integer part is 3, and x

can be written as:

337 89 = 3 + 70 89 = 3 + 1 89 70 ;

now consider the reciprocal of the fractional part of x, i.e., 89₇₀. Its integer part is 1, and we can write

89

70 = 1 + 19 70.

We carry on the procedure inductively, until we obtain the following expres-sion of x = 337₈₉ as a continued fraction:

337 89 = 3 + 1 1 + 1 3 + 1 1 + 1 2 + 1 6 Thus, E(337, 89) = 6.

The next computation clarifies the connection with the Euclidean algorithm: we successively integer divide the denominator of the fractional part for its

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numerator, until the remainder of the division equals zero. 337 = 3 · 89 + 70 89 = 1 · 70 + 19 70 = 3 · 19 + 13 19 = 1 · 13 + 6 13 = 2 · 6 + 1 6 = 6 · 1 + 0.

Matveev’s formula for the complexity of a lens space

Here we shall state the upper bound on complexity of lens spaces established by Matveev in 1990, see [Matb], and advertised in the introduction. Let us denote by c(L(p, q)) the complexity of the lens space L(p, q); without loss of generality, we can assume that 0 ≤ q < p. The conjectural formula established by Matveev is the following:

c(L(p, q)) = E(p, q) − 3. (1.1)

Matveev actually proved the inequality

c(L(p, q)) ≤ E(p, q) − 3.

This task was accomplished by explicitly constructing a special spine of L(p, q) with E(p, q) − 3 vertices. In the following, we shall briefly describe how a special spine having this number of vertices can be built.

The construction of the conjectured minimal spine of L(p, q) is based on a suitable encoding of the slopes on the boundary of the solid torus. This encoding uses the so-called Farey tessellation of the hyperbolic plane, and its dual graph. In the following, we shall briefly describe the construction of the spine. However, further details can be found in [MP04]. The main idea is as follows: since a lens space can be viewed as the gluing along their boundary of two solid tori, one would like to build a spine of a lens space by gluing two spines of the solid torus. Nevertheless, a direct gluing of the two spines may give rise to a polyhedron which is not special. For this reason, the gluing have to be factorized in more steps, where each of these steps consist in adding a so-called elementary brick to the spine.

The Farey tessellation and its dual graph

Let us denote by S(T ) the set of the slopes on the boundary of the solid torus, i.e., of the classes of isotopy of the non-trivial simple closed curves on

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1/2 3/2 −1 −2 −1/2 1 2 3 8 0 1/3 2/3

Figure 1.23: The Farey tessellation of the hyperbolic plane.

T . Fix a basis (µ, λ) for H1_{(T ). Then , there is a bijection}

Φ : S(T ) −→ Q ∪ {∞},

where Φ(γ) = p/q for γ = ±(pµ + qλ). Let us consider the half-plane model of the hyperbolic plane H2_{: then we can think Q ∪ {∞} as a subset of}

∂H2 = R ∪ {∞}. By abuse of terminology, we will denote Q ∪ {∞} by ∂QH 2_.

Definition 1.6.9. The Farey tessellation of the hyperbolic plane is obtained by joining with a line two points Φ(γ) and Φ(γ0) in ∂_Q_H2 _{if the two curves}

γ and γ0 intersect exactly once. Figure 1.23 shows the Farey tessellation, drawn in the disc model of the hyperbolic plane.

By considering the half-plane model for the hyperbolic plane, it turns out that there exists a line joining two rational numbers p/q and s/t if and only if |p·t−q ·s| = 1. Hence, the Farey tessellation actually does not depend on the choice of the basis. Now denote by Θ(T ) the set of isotopy classes of θ-graphs in T having a disc as a complement. One easily observes that there exists a bijection between the elements of Θ(T ) and the triples of slopes having pairwise intersection numbers ±1. Indeed, three slopes as above are exactly those contained in θ. Hence, by the definition of the Farey tessellation follows that a triple of slopes {γ, γ0, γ00} defines a θ-graph if and only if Φ(γ), Φ(γ0₎

and Φ(γ000) are the vertices of a piece of the Farey tessellation. Now denote by G the graph dual to the Farey tessellation, see Figure 1.24.

Then there exists a bijection

Ψ : Θ(T ) −→ G(0)

sending a θ-graph in the centre of the triangle of the tessellation whose sides corresponds to the slopes in θ. Notice that G is trivalent, hence there exist exactly three θ-graphs linked to a fixed θ ∈ Θ by a unique edge of G. More-over, these three θ-graphs are those which can be obtained from θ by a flip

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−1/2 1/2 1 3/2 2 3 8 −2 −1 0 1/3 2/3

Figure 1.24: The graph G, dual to the Farey tessellation.

Figure 1.25: The two types allowed for each component of the intersection of the embedded normal surface F with each tetrahedron.

as shown in Figure 1.25. We can define a sort of distance between slopes and θ-graphs.

Definition 1.6.10. Let be γ ∈ S(T ) and θ ∈ Θ(T ). We define d(γ, θ) = n − 1, where n is the number of sides of the Farey tessellation intersected by the half line in H2 joining the two vertices Ψ(θ), Φ(γ) ∈ G(0).

Definition 1.6.11. Denote by B2 the solid torus equipped whit a θ-graph as

shown in Figure 1.26. We will denote by θ the θ-graph in B2: θ corresponds

via the bijection Ψ to the centre of the triangle having vertices {0, 1, 1/2}. It turns out —see [MP04] for full details— that we can construct a special spine for the lens space L(p, q) having as much vertices as d(p/q, θ), which

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Figure 1.27: The elementary brick.

equals E(p, q) − 3. Denote by r the half line in the hyperbolic plane joining the points Φ(γ) with Ψ(θ). The special spine mentioned above can be built starting with B2 and adding a brick of the type shown in Figure 1.27 for each

intersection of r with the sides of the Farey tessellation.

1.7 Basics of Haken’s theory

of normal surfaces

This last section is aimed at briefly introducing another tool that we will often use in the subsequent chapters, Haken’s theory of normal surfaces.

The central idea of the beautiful theory of Haken is that one can investi-gate the properties of a 3-manifold looking at some particular surfaces, called normal surfaces, contained in it. A normal surface will be defined as a sur-face that admits a representative —up to isotopy— which can be built as an assembly of some preassigned elementary pieces. Further details will be given later on.

Let M be a closed, connected 3-manifold with a fixed triangulation T , and denote by T(1) _{the 1-skeleton of T . Let F ⊂ M be a closed, embedded}

surface. Up to isotopy, we may assume that F is transverse to T , i.e., F does not intersect any vertex, intersects transversely T(1) in a finite number of points and its intersection with each 2-simplex consists in a finite union of simple closed curves and arcs.

Now we can introduce the fundamental notion of normal surface.

Definition 1.7.1. Let M be a closed, connected manifold with a fixed tri-angulation T . An embedded surface F ⊂ M is called normal with respect to T if for each tetrahedron ∆ of T each component of F ∩ ∆ is a disc of one

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Figure 1.29: An example of the intersection between a normal surface and a tetrahedron of the triangulation.

of the two types shown in Figure 1.28. We will call a disc of the first type a triangle, of the second type a quadrilateral, or simply a quad.

Figure 1.28: The two types allowed for each component of the intersection of the embedded normal surface F with each tetrahedron.

Notice that each triangle cuts off a vertex, while a quad separates a pair of opposite edges. Hence, for each tetrahedron ∆ of T each component of the intersection F ∩ ∆ is one of 7 possible types: there are 4 triangle types, one for each vertex of ∆, and 3 quad types, as much as the pairs of opposite edges. Figure 1.29 shows an example of the way how a normal surface can intersect a tetrahedron of T .

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Chapter 2 Layered triangulations of solid

tori and lens spaces

The main purpose of this chapter is to supply the reader with an introduc-tion to the theory of layered triangulaintroduc-tions of solid tori and lens spaces. This theory appears for the first time in [JR06]. The fundamental definitions will be given, as well as a way to simply encode this particular family of triangu-lations.

2.1 Triangulations and layering on an edge

Through the entire dissertation we will use the term ”triangulation” meaning what is more commonly encountered in literature as ”singular triangulation”. We now introduce a definition in order to establish once and forever what we mean.

Definition 2.1.1. Let ∆ := ˜∆1t . . . t ˜∆n be a disjoint union of tetrahedra.

We call face-pairing on ∆ a family Φ of affine homeomorphisms between facets of the ˜∆i’s, such that φ ∈ Φ if and only if φ−1 ∈ Φ, and each facet

of each ∆i is the domain of exactly one φ ∈ Φ. We call partial face-pairing

a family Φ of affine homeomorphism between facets of the ˜∆i’s, such that

φ ∈ Φ if and only if φ−1 ∈ Φ, and each ∆i is the domain of at most one

φ ∈ Φ.

A (partial) family of face-pairings induces an equivalence relation on ∆. We will denote by ∆/Φ the quotient space with respect to this equivalence

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relation, and by p : ∆ −→ ∆/Φ the associated projection. A priori the resulting polyhedron is not a 3-manifold; but if we require p to be injective on the interior part of every edge we can obtain a 3-manifold except possibly at the image of some vertices. Indeed, under this assumptions, after that the gluing has been performed the link on any point other than a vertex is a circle. The notion of triangulation of a surface is easily given along the same lines.

Definition 2.1.2. Suppose a 3-manifold M is homeomorphic to ∆/Φ, where the associated map p : ∆ −→ ∆/Φ is injective on the interior part of every edge; then we set T := (∆, Φ) and say T is a triangulation of M . We will call vertex, edge, triangle and tetrahedron the image in M of a j-simplex of ∆ (with j = 0, 1, 2, 3 respectively) under the projection p : ∆/Φ −→ M. Definition 2.1.3. Suppose that e is an edge of a triangulation T = (∆, Φ) of a 3-manifold M . We call degree of e the number of edges ˜e of tetrahedra in ∆ that are mapped on e by the projection p.

It is known that every closed 3-manifold admits a one-vertex triangula-tion; this result will be crucial in the following. Now we define an operation that will play a fundamental role in the sequel. Suppose M is a compact 3-manifold with non-empty boundary and T = (∆, Φ) is a triangulation of M that restricts to a triangulation P of ∂M ; select a boundary edge e in P such that there are two distinct triangles σ and β meeting in e. Let ˜∆ be a tetrahedron not belonging to ∆, choose one of the edges ˜e of ˜∆ and denote by ˜

σ e ˜β the two faces of ˜∆ containing ˜e. Now identify ˜e with e, and also extend the identification between the faces of the same name; this way we obtain a new triangulation T0 of M such that the induced triangulation, say P0, of ∂M differs from P for what we call a diagonal flip in the quadrilateral σ ∪ β, see Figure 2.1. There are two ways to perform the gluing of ˜∆; however, the two resulting triangulations are isomorphic.

Definition 2.1.4. We call this operation a layering on the triangulation T along the edge e.

2.2 Layered triangulations of solid tori

In this paragraph we shall introduce a particular family of triangulations of the solid torus. We will start with a brief discussion about the one-vertex triangulations of the torus.

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Figure 2.1: Layering on T along e.

Figure 2.2: One-vertex triangulation of the torus. It is unique up to homeomor-phism, but not up to isotopy.

2.2.1 One-vertex triangulations of the torus

Suppose P is a one-vertex triangulations of the torus, as shown in Figure 2.2. It follows from an Euler characteristic argument that such a triangulation has two faces and three edges, say e1, e2, e3. If γ is an isotopy class of simple,

closed, normal curves, we can associate to it a triple of non-negative integers {y1, y2, y3} = {|γ ∩ e1|, |γ ∩ e2|, |γ ∩ e3|}, where |γ ∩ ei| denotes the number

of times the curve γ intersects the edge ei of the triangulation.

Lemma 2.2.1. If γ is essential, then relation yk= yi+yj holds for a suitable

choice of {i, j, k} = {1, 2, 3}.

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Figure 2.3: The three edges e1, e2 and e3, and the numbers of intersection of γ

with them.

essential. Let us consider Figure 2.3. The following relation holds: yi = X k6=i nj,k, so we have yi1 + yi2 = yi3 + 2nj,i3.

On the other hand there exists (j, i) such that nj,i = 0, otherwise γ would be

the link of the vertex.

Suppose γ represents a slope on the torus, i.e., an isotopy class of non-trivial simple closed curves; then it determines a triple {y1, y2, y3}. This

applies in particular to a special slope on the boundary of a solid torus, which we will refer to as the meridional slope, characterized by the property that any curve that represents such a slope bounds a disk in the solid torus. Suppose T is a triangulation of the solid torus that determines a one-vertex triangulation T∂ of the boundary; by considering the intersection

numbers between the meridional slope and the edges, we obtain a unique associated triple {p, q, p + q}, with p ≤ q. Observe that the triples {0, 1, 1} and {1, 1, 2} are allowed: they correspond respectively to the case of the meridional slope being the slope of an edge (see Figure 2.4) and to the case shown in Figure 2.5.

Except for the two cases mentioned above, we may assume 1 ≤ p < q and p, q relatively prime; furthermore, the triple {p, q, p + q} is completely determined by p and q. This allows us to introduce the following:

Definition 2.2.2. We will call p/q-triangulation a one-vertex triangulation of the boundary of the solid torus whose associated triple with respect to the meridional slope is {p, q, p + q}.

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Figure 2.4: The triangulation of the solid torus corresponding to the triple {0, 1, 1}.

Figure 2.5: The triangulation of the solid torus corresponding to the triple {1, 1, 2}.

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The following lemma states that the rational number p/q in [0, 1] uniquely determines a one-vertex triangulation of the solid torus, up to homeomor-phism.

Lemma 2.2.3. Let T∂ and T∂0 be p/q and p

0_/q0 _{triangulations, respectively,}

on the boundary of the solid torus T. Then there exists an automorphism of T taking T to T0 if and only if p/q = p0/q0.

Proof. The rational number p/q is clearly unique up to homeomorphism. We now prove that p, q ∈ N with p ≤ q and (p, q) = 1 uniquely determine a p/q-triangulation of ∂T up to automorphism of T. To this end we choose a basis (µ, λ) of H1(∂T) with µ a meridian of T, and we recall that an

automorphism Φ of T is uniquely determined up to isotopy by its trace on ∂T, which in turn is determined by the matrix [Φ] of its action on H1(∂T)

with respect to (µ, λ). Moreover, the possible matrices [Φ] are precisely those of the form: −1 0 0 1 ε1 · 1 0 0 −1 ε2 · 1 1 0 1 k , (2.1)

with ε1, ε2 ∈ {0, 1} and k ∈ Z. Notice that

−1 0 0 1 and 1 0 0 −1

represent reflections, while

1 1 0 1

represents a meridional Dehn twist.

Next, we recall that a simple closed non-trivial curve on ∂T is determined up to isotopy by its image up to sign in H1(∂T), or more precisely that it

can be described as ±(cµ + dλ) for some c, d ∈ Z with (c, d) = 1.

We are now ready to show that there exists and is unique up to isotopy of T a triple α, β, γ of curves that intersect pairwise transversely at one and the same point, giving a p/q-triangulation of ∂T. To begin with, we must impose that |α t µ| = p and |β t µ| = q, so we must have α = ±(aµ + pλ) and β = ±(bµ + qλ). Next, we must impose that |α t β| = 1, which means aq − bp = ±1. Since (p, q) = 1, two integers a and b satisfying this relation exist; they are not unique, though, as a given pair a, b can be replaced by a0, b0 with:

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(

a0 = ± a + hp

b0 = ± b + hq. (2.2)

However, we can show that the lack of uniqueness of a, b is compensated by the action of the automorphisms of T, so the pair of curves (α, β) is unique up to this action. Indeed, if we compute the images of the two curves α = ±(aµ + pλ) and β = ±(bµ + qλ) via an automorphism of the solid torus Φ whose matrix with respect to the basis (µ, λ) is as in Equation (2.1), we obtain α0 = ±(a + kp)µ ± pλ, β0 = ±(b + kq)µ ± q. Whence, a0 = ±(a + kp), b0 = ±(b + kq)

consistently whit Equations 2.2. Having shown that up to the automorphisms of T we have unique curves α, β with |α t µ| = p, |β t µ| = q and |α t β| = 1, we note that the complement of α ∪ β in ∂T can be viewed as a square, and for precisely one diagonal γ of this square we have |γ t µ| = p + q. The triple α, β, γ thus found, which exists and is unique up to isotopy of T, gives the desired p/q-triangulation.

2.2.2 Layered triangulations of the solid torus:

defini-tion

In this section we shall define a family of quite special triangulations of a solid torus. Before formalizing what we will call a layered triangulation, we provide the reader with an informal but effective idea of these objects. Let T be a one-vertex triangulation of a solid torus T, and let e be one of the three edges on ∂T; then using the technique of layering along the edge e presented in the previous section, we can add a new tetrahedron ˜∆ to T . The resulting manifold is still a solid torus, equipped with a new triangulation which has one more tetrahedron than T .

Definition 2.2.4. We call one-triangle M¨obius strip the surface obtained as the quotient space of a triangle via the identification of two of its sides shown in Figure 2.6. Notice that this way we obtain a one-triangle triangulation

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Figure 2.6: The one-triangle M¨obius strip

Figure 2.7: Folding along the edge e, or layering along e onto the one-triangle M¨obius strip.

of the M¨obius strip having one vertex, one boundary edge, and one interior edge.

Definition 2.2.5. We call the operation described in Figure 2.7 folding along the edge e. In the sequel, by abuse of terminology we will also refer to this operation as to a layering on the one-triangle M¨obius strip.

To begin with, we are interested in understanding what can we obtain by layering a tetrahedron on the one-triangle M¨obius strip. The result of this operation is shown in Figure 2.8. The back two faces give the M¨obius strip,

A

B

C

D

_ABC=DAB

Figure 2.8: The one-vertex triangulation of the solid torus; the two back faces give rise to the one-triangle M¨obius strip.

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Figure 2.9: The one-triangle M¨obius strip, represented inside the solid torus. It corresponds to the two back faces of the tetrahedron in Figure 2.8. and the resulting total space is the solid torus with all edges on the boundary but te rest of the M¨obius strip inside.

To visualize this, start with the M¨obius strip inside the solid torus, see Figure 2.9. Now we push the core curve of the strip to the boundary, nor-mally, except near the vertex, see Figure 2.10. The two curves meeting at

Figure 2.10: The solid torus, the one-vertex M¨obius strip inside it, and the core curve of the M¨obius strip, pushed on the boundary.

the vertex determine a square in ∂T, and the third edge e3 will be one of the

two diagonals. Choosing representative for the meridian µ and the longitude λ as shown in Figure 2.11, we have:

e1 = λ + µ,

e2 = −2λ − µ.

Now we shall write the third edge e3 as a linear combination of µ and λ;

as a diagonal of the square having e1 and e2 as its side, there are only four

possibilities: ±λ and ±(3λ + 2µ). In order to identify e3, we compute the

fundamental group from scratch; it is generated by e1, e2 and e3, and the

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Figure 2.11: Representatives for the meridian and the longitude of the solid torus. π1 =e1, e2, e3| e21· e2, e1· e−12 · e −1 3 , e1 · e−13 · e −1 2 . (2.3) Whence we have π1 = he1i ,

and the following relations hold:

e2 = e−21 ,

e3 = e31.

So we have that e3 is 3λ + 2µ; Figure 2.12 shows the resulting one-vertex

triangulation of the solid torus. The computation above also allows us to

de-Figure 2.12: The one-vertex triangulation of the solid torus.

scribe the meridional disc of the solid torus as a normal surface with respect to the 1-layered triangulation. To begin with, let us visualize the meridional disc inside the solid torus, see Figure 2.13. In this figure, the intersections between the meridional disc and the edges of the one-vertex triangulation of the solid torus are shown. In particular, the disc is divided by the edges in three regions; we will refer to the intersection between the boundary cir-cle of the meridional disc and one of the edges as to a vertex. Using this terminology, two of the regions —labelled T1 and T2 in Figure 2.13— have

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Figure 2.13: The meridional disc of the solid torus equipped with the one-vertex triangulation.

three vertices, while the third region — labelled Q— has four vertices in its boundary. The regions correspond respectively to two normal triangles, T1 and T2, and to a quadrilateral, Q, in the one-vertex triangulation; the

meridional disc as a normal surface respect to the one-vertex triangulation is shown in Figure 2.14.

Figure 2.14: The meridional disc of the solid torus as a normal surface with respect to the one-vertex triangulation.

Now let us formalize the definition of a layered triangulation of T. Definition 2.2.6. We inductively define for t ≥ 1 a t-layered triangulation of the solid torus as the result of one layering on:

• for t = 1, the one-triangle M¨obius band.

• for t ≥ 2, a (t − 1)-layered triangulation of the solid torus.

By abuse of terminology, we will refer to the one-triangle M¨obius band as a 0-layered triangulation of the solid torus.

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Observation. A layered triangulation of a solid torus with t layers has one vertex, which is in the boundary torus, t + 2 edges, three of which are located in the boundary, and 2t + 1 faces, two of which are in the boundary torus. Notice that in any layered solid torus there is a special edge in the boundary having degree one: it is called the univalent edge.

By Lemma 2.2.3 there exists a one-to-one correspondence between the rational numbers p/q in [0, 1] and the equivalence classes up to homeomor-phism of the solid torus of one-vertex triangulations of the boundary. Given a one-vertex triangulation of the boundary, the associated rational number is determined by the number of intersections of the boundary edges with a curve representing the meridional slope. We are interested in how the ra-tional number changes after performing a layering on a boundary edge (this operation clearly changes the boundary triangulation, and consequently the intersection numbers between the meridional slope and the new boundary edges).

Lemma 2.2.7. Let T be a triangulation of T inducing the p/q-triangulation of ∂T. The three possible layering on T induce the p0/q0_{-triangulation of ∂T,} where p0/q0 ranges in p p + q, q p + q, min p q − p, q − p p (2.4) Proof. In Figure 2.15 the effects of each possible layering on a boundary edge are shown. In each case, α, β and α + β are the curves representing the edges of the p/q triangulation. For each set of three relations in Figure 2.15, the first one describes how the new set of curves — corresponding to the edges of the new p0/q0 triangulation— is related to the old one. In the second line, the slopes of the new and old edges are reported. Finally, in the third line the rational number p0/q0 is expressed as a function of p and q.

We shall build a graph that will encode how to construct a p/q-layered triangulation starting from the one-vertex triangulation of the solid torus. Definition 2.2.8. We call L-graph the graph whose vertices are the rational numbers in [0,1], with an edge joining two distinct vertices p/q to p0/q0 if and only if p0/q0 appears in Equation 2.4, see Figure 2.16.

Notice that by specifying that there is an edge connecting two distinct vertices we are taking into consideration all the possible relation of one the types mentioned in Equation 2.4, except for the relation of 0/1 with itself.

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Figure 2.16: A portion of the L-graph. Proposition 2.2.9. The L-graph is a tree.

Proof. Define h(p_q) = q, and observe that for each vertex other then 0/1 there exists exactly one edge departing from it along which h decreases, while h increases along the remaining edges, that are in number of two but for the vertices 0/1 and 1/1, for which there exists only one edge with this property. If a simple simplicial closed non-trivial path were to exist, by choosing its vertex realizing the minimum for h we would get a contradiction; hence the L-graph contains no cycles, and it is a tree.

Theorem 2.2.10. Every one-vertex triangulation of ∂T is the boundary trace of a layered triangulation of T.

Proof. In a topological language, this amounts to saying that the 0/1 vertex of the L-graph is connected to any other p/q vertex; on the other hand, in an arithmetic language it amount to saying that one can go from 0/1 to any other rational number p/q using moves of the form

a b −→ a a + b or b a + b.

The latter fact easily follows from the Euclidean algorithm.

To each vertex of the L-graph by Lemma 2.2.3 there corresponds a one-vertex triangulation of the boundary of the solid torus, unique up to home-omorphism. The next proposition describes how to construct a p/q-layered triangulation of the solid torus, by following a path on the L-graph. The proof is obvious by construction.

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Proposition 2.2.11. The paths in the L-graph starting at p/q and ending at 1/1 are in one-to-one correspondence with the layered triangulations of the solid torus extending the p/q triangulation on the boundary. Furthermore, any arc in the path corresponds to the layering of a new tetrahedron, so that the number of edges forming the path equals the number of tetrahedra in the corresponding triangulation.

Definition 2.2.12. A p/q-layered triangulation of the solid torus T is called minimal if it corresponds to a minimal path in the L-graph joining p/q with 1/1.

Corollary 2.2.13. There exists a unique minimal layered triangulation of the solid torus extending the p/q-triangulation of its boundary.

In the sequel we will be especially interested in a specific family of layered triangulations of the solid torus, that we will now describe.

2.2.3 A particular family of triangulations

Given k ∈ N with k ≥ 1 we now describe combinatorially the minimal layered triangulation of the solid torus extending the {1, k + 1, k + 2} triangulation of its boundary; furthermore, we analyse the degree of each edge in such a triangulation.

Let S1 be the 1-layered triangulation of the solid torus, see Figure 2.8.

Now a layering of a tetrahedron ∆2 on the edge e2 is performed: denote

by S2 the resulting solid torus. In S2 there exists a unique edge, say e4,

which is not identified with any other previous edge; we orient e4 so that its

incident faces give rise to the relation e4 = e1+e3. We carry on the procedure

inductively; at the step k we layer a new tetrahedron ∆k on the edge ek and

we orient the new edge ek+2 so that it fulfills the relation ek+2 = e1 + ek+1,

see Figure 2.17. We denote the resulting triangulated solid torus by Sk.

Identifying the fundamental group of the solid torus π1(T) with Z and

bearing in mind the relation given from faces, we obtain [ek] = k ∈ Z.

According to their corresponding integer in Z, edges in Sk are termed odd

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Figure 2.17: A description of how the boundary triangulation of Sk−1

changes after a layering on ek.

2.2.4 An analysis of edge degrees in the

layered triangulation

S

n

of a solid torus

Let ek be an edge in a layered triangulation of the solid torus; ek appears for

the first time at a precise stage with a certain degree, to which we will refer as birth degree and denote by db(ek). In the following we will refer to a stage as

to the layering of a new tetrahedron, so that the number of the stages equals the number of tetrahedra. Calling e1, e2, e3 the three edges in the one-vertex

triangulation of the solid torus and denoting by ei with increasing indices i

every new edge introduced in the layering procedure, we have: db(ek) =

(

4 − k if k ∈ {1, 2, 3} 1 otherwise.

Once the edge ek appears for the first time in a layered triangulation, it is a

boundary edge and it remains a boundary edge until we layer on it for the first time. Notice that before we perform a layering on ek, its degree increases

by two at any new step. When we layer on ek its degree increases by one,

and later the degree of ek does not change any longer, because ek becomes

an interior edge. Denote by lb(ek) the stage at which ek appears for the first

time; we have

lb(ek) =

(

1 if k ∈ {1, 2, 3} k − 3 otherwise.

Denote by (ei1, . . . , ein) the n-tuple which parameterizes the way how the layering has to be performed, i.e., the n-tuple has edge ei in position j if at

step j we layer on ei. Notice that in order for the n-tuple to be admissible,

i.e., to actually describe a sequence of layerings, each ej can appear just once,

due to the fact that after a layering on a boundary edge the edge becomes an interior one, and further layerings on it are not allowed; furthermore, if the edge ei is located in position j, then i ≤ j + 2.

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Denote by dj(ek) the degree of ek at stage j, where j ≥ lb(ek). Then the

following formula holds:

dj(ek) =

(

db(ek) + 2((s − 1) − lb(ek)) + 1 if ∃ s < j such that eis = ek db(ek) + 2(j − lb(ek)) otherwise.

The process can be easily visualized in the next table, where the edge degrees of the layered triangulations Sk for 1 ≤ k ≤ 4 are shown. We put a star

in the box corresponding to an edge affected twice by a layering, we under-line numbers located in boxes corresponding to those edges along which we perform the layering, and we drawn a square in the boxes corresponding to edges which have become internal, and whose degree will not change.

e1 e2 e3 e4 e5 e6

S1 3 * 2 1 *

S2 5 * 3 3 1 *

S3 7 * 3 4 3 1*

S4 9 * 3 4 4 3 1 *

To sum up, the next table shows the degree of edges in Sk.

edge e1 e2 e3 to ek ek+1 ek+2

degree 2k+1 3 4 3 1

2.3 Layered triangulations of lens spaces

In this section we shall define layered triangulations of lens spaces; in order to accomplish this task, we will exploit the genus-one Heegard decomposition of lens spaces, together with the definition of layered triangulations for solid tori which has been introduced in the previous section.

Definition 2.3.1. We call layered triangulation of a lens space a triangula-tion obtained by gluing together two triangulatriangula-tions of the solid torus under a map matching their boundary traces.

Remark 1. Suppose a layered triangulation T of some lens space L(p, q) is obtained by gluing triangulations T1 of T1 and T2 of T2. Suppose

fur-thermore that Tj is a kj-layered triangulation of Tj consisting of tetrahedra

∆(j)₁ , . . . , ∆(j)_k

j, appearing in this order during the layering. Then the layered triangulation T admits the following equivalent description:

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- Use ∆(1)₁ , . . . , ∆(1)_k

j in order to construct T1.

- Use the attaching map of T1 and T2 to layer the tetrahedron ∆ (2) k2 on the triangulation already constructed.

- Proceed by successively layering ∆(2)_k

2−1, . . . , ∆

(2)

1 , thus eventually

get-ting a (k1+ k2)-layered triangulation ˜T of the solid torus T.

- Perform a folding along one of the boundary edges of ˜T .

Definition 2.3.2. A layered triangulation of a lens space is called minimal if it has the minimum number of tetrahedra among all the possible layered triangulation of the space under consideration.

We state and prove a result that will be useful in the sequel. Theorem 2.3.3. Every lens space admits a layered triangulation.

Proof. Take a genus-one Heegard splitting of the lens space, i.e., realize it as T1∪ T2 where T1, T2 are solid tori, and T1∩ T2 = ∂T1 = ∂T2 =: T.

Fix a one-vertex triangulation of the splitting torus T ; by Theorem 2.2.10, the boundary triangulation extends to a layered triangulation of the solid torus on each side, giving rise to two layered triangulation of the solid tori T1and T2 whose attaching gives the required layered triangulation of M .

Now we are interested in how to infer the parameters of a lens space obtained by performing a folding along one of the boundary edges of a p/q-layered triangulation of the solid torus. One way to accomplish this task is the following:

1. Without loss of generality, we can assume that the triangulation is minimal. Hence we can look at it as obtained by layering a unique further tetrahedron on the r/s triangulation of the solid torus, where

(

r = p, s = q − p if q ≥ 2p r = q − p, s = p if q ≤ 2p

2. Denoting by µ and λ the meridian and the longitude of the solid torus respectively, we write the boundary edges of the r/s-triangulation as

ε1 = aµ + rλ,

ε2 = bµ + sλ,

Matveev complexity of some lens spaces

Introduction

Contents

Chapter 1

Preliminary notions

1.1

A primer on PL-Topology

1.1.1

Collapse

1.1.2

Spines

A

C

B

C

D

D

1.2

Complexity of 3-manifolds

1.2.1

Almost simple polyhedra

1.3

The complexity function

1.4

Properties of complexity

A

1.5

Non-minimality criteria

P

1.6

Lens spaces and their complexity

1.6.1

Lens spaces: definition and properties

v

v

v

1.6.2

The conjectural formula

1.7

Basics of Haken’s theory

of normal surfaces

Chapter 2

Layered triangulations of solid

tori and lens spaces

2.1

Triangulations and layering on an edge

2.2

Layered triangulations of solid tori

2.2.1

One-vertex triangulations of the torus

2.2.2

Layered triangulations of the solid torus:

defini-tion

A

B

C

D

ABC=DAB

2.2.3

A particular family of triangulations

2.2.4

An analysis of edge degrees in the

layered triangulation

S

of a solid torus

2.3

Layered triangulations of lens spaces

_C

_ABC=DAB