A complexity of compact 3-manifold via immersed surfaces
aa r X i v : . [ m a t h . G T ] F e b A complexity of compact 3-manifold viaimmersed surfaces
Gennaro Amendola
February 12, 2021
Abstract
We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a naturalnumber measuring how much the manifold is complicated. We prove thatit fulfils interesting properties: it is subadditive under connected sum andfinite-to-one on P -irreducible and boundary-irreducible manifolds with-out essential annuli and Möbius strips. Moreover, for these manifolds,it equals the minimal number of cubes in a cubulation of the manifold,except for the sphere, the ball, the projective space and the lens space L , , which have surface-complexity zero. We will also give estimationsof the surface-complexity by means of ideal triangulations and Matveevcomplexity. Introduction
The problem of filtering compact 3-manifolds in order to study them system-atically has been approached by many mathematicians. The aim is to find afunction from the set of compact 3-manifolds to the set of numbers. The numberassociated to a 3-manifold should be a measure of how much the manifold iscomplicated. For instance, for compact surfaces, this can be achieved by meansof genus. For compact 3-manifolds, many possible functions have been found:e.g. the Heegaard genus, the Gromov norm, the Matveev complexity.Each of these functions have features that can be used to study (classes of)compact 3-manifolds. For instance, they are additive under connected sum.However, some of them have drawbacks. The Heegaard genus and the Gromovnorm are not finite-to-one, while the Matveev complexity is. Hence, in orderto carry out a classification process, the latter one is more suitable than theformer ones. The Matveev complexity is also a natural measure of how muchthe manifold is complicated, because if a 3-manifold is P -irreducible, boundary-irreducible, without essential annuli and Möbius strips, and different from theball B , the sphere S , the projective space RP and the Lens space L , , thenits Matveev complexity is the minimal number of tetrahedra in an ideal trian-gulation of its (the Matveev complexity of B , S , RP and L , is ). Suchfunctions could also be tools to give proofs by induction. For instance, the1eegaard genus was used by Rourke to prove by induction that every closedorientable 3-manifold is the boundary of a compact orientable 4-manifold [19].For closed 3-manifolds, one of this function is the surface-complexity, de-fined in [3] by means of triple points of images of transverse immersions ofclosed surfaces that divide the manifold into balls. It is a function from the setof closed 3-manifolds to the set of natural numbers, which fulfils some prop-erties: it is subadditive under connected sum, and, in the P -irreducible case,it is finite-to-one and it equals the minimal number of cubes in a cubulationof the manifold. Analogous interesting definitions (which inspired the surface-complexity one) are the Montesinos complexity and the triple point spectrum,given by Vigara [22] and studied by Lozano and Vigara [11, 12, 13]. The defi-nition of surface-complexity is similar to that of Montesinos complexity, but ithas the advantage of being more flexible, allowing to prove the properties listedabove.The aim of this paper is to generalise the definition of surface-complexityto the compact case, to generalise the properties holding in the closed case andto give bounds. We plan to give a complete list of compact 3-manifolds withsurface-complexity one in a subsequent paper, as done for the orientable closedcase in [4], see also [10].We now sketch out the definition and the results of this paper. The surface-complexity of a connected and compact 3-manifold will be defined by means ofthe notion of quasi-filling Dehn surface , i.e. the image of a transverse immersionof a closed surface to which the manifold, possibly after removing some opendisjoint balls, collapses. This generalises the notion of quasi-filling Dehn surfacegiven for closed 3-manifolds in [3], which in the end turns out to be a weakerversion of the notion of filling Dehn surface given by Montesinos [17]. It isworth noting that (quasi-)filling Dehn surfaces are not yet studied as the otherstructures (e.g. Heegaard splittings, triangulations, spines) used to define theother functions described above (there are only partial results [22, 2, 23, 12]). Definition
The surface-complexity sc ( M ) of a connected and compact 3-manifold M is the minimal number of triple points of a quasi-filling Dehn surface of M .For the closed case, this notion differs from Montesinos complexity in two points:firstly quasi-fillingness is used instead of fillingness, secondly any closed surfaceis allowed as the domain of the transverse immersion instead of the sphereonly. Therefore the surface-complexity could be also called weak-Montesinoscomplexity , and for closed 3-manifolds the inequality sc ( M ) mc ( M ) obviouslyholds, where mc ( M ) denotes the Montesinos complexity of M . In this paper,we have decided to stick to “surface-complexity” to stress the similarities withthe closed case studied in [3].We will prove the following three properties. Finiteness
For any integer c there exists only a finite number of connected, compact, P -irreducible and boundary-irreducible 3-manifolds without essential an-nuli and Möbius strips that have surface-complexity c . Naturalness
The surface-complexity of a connected, compact, P -irreducible and boundary-irreducible 3-manifold without essential annuli and Möbius strips, differentfrom S , B , RP and L , , is equal to the minimal number of cubes in an2deal cubulation of M . The surface-complexity of S , B , RP and L , iszero. Subadditivity
The surface-complexity of the connected sum and of the boundary con-nected sum of connected and compact 3-manifolds is less than or equal tothe sum of their surface-complexities.The naturalness property will follow from the features of minimal quasi-fillingDehn surfaces of connected, compact, P -irreducible and boundary-irreducible3-manifolds without essential annuli and Möbius strips, where minimal means“with a minimal number of triple points”. A quasi-filling Dehn surface of a con-nected and compact 3-manifold is called filling if it is cell-decomposed by itssingularities. The cell-decomposition dual to a filling Dehn-surface is actuallyan ideal cubulation of the manifold. Hence, in order to prove the natural-ness property, we will prove that every connected, compact, P -irreducible andboundary-irreducible 3-manifold without essential annuli and Möbius strips, dif-ferent from S , B , RP and L , , has a minimal filling Dehn surface. We pointout that not all minimal quasi-filling Dehn surfaces of connected, compact, P -irreducible and boundary-irreducible 3-manifolds without essential annuli andMöbius strips are indeed filling. However, they can be all constructed startingfrom filling ones (except for S , B , RP and L , , for which non-filling onesmust be used) and applying a simple move, we will call bubble-move .The surface-complexity is related to the Matveev complexity. Indeed, if M is a connected, compact, P -irreducible and boundary-irreducible 3-manifoldwithout essential annuli and Möbius strips, different from L , and L , , thedouble inequality c ( M ) sc ( M ) c ( M ) holds, where c ( M ) denotes theMatveev complexity of M .The two inequalities above give also estimates of the surface-complexity. Ingeneral, an exact calculation of the surface-complexity is very difficult, howeverit is relatively easy to estimate it. More precisely, it is quite easy to give upperbounds for it, because constructing a quasi-filling Dehn surface of the manifoldwith the appropriate number of triple points suffices. With this technique, wewill give an upper bound for the surface-complexity of a connected and compact3-manifold starting from an ideal triangulation of its. The problem of provingthe sharpness of this bound is usually difficult (see for instance [13] where thetriple point spectrum of S and S × S has been computed very subtly).In the Appendix we will give a brief description of what happens in the 2-dimensional case. We plan to cope with the 4-dimensional case in a subsequentpaper. Acknowledgements
We would like to thank Bruno Martelli for the usefuldiscussions on the Matveev complexity.
Throughout this paper, all 3-manifolds are assumed to be connected and com-pact. By M , we will always denote such a (connected and compact) 3-manifold.Using the Hauptvermutung , we will freely intermingle the differentiable, piece-wise linear and topological viewpoints.3 implepoint Doublepoint Triplepoint
Figure 1: Neighbourhoods of points (marked by thick dots) of a Dehn surface.The manifold M is called P -irreducible if every sphere in M bounds a balland M does not contain any two-sided embedded projective plane. The manifold M is called boundary-irreducible if for every proper disc D ⊂ M the curve ∂D bounds a disc in ∂M . Suppose M is P -irreducible and boundary-irreducible,then a proper annulus A ⊂ M is called essential if A does not cut off from M ei-ther a cylinder (a ball intersecting ∂M in two discs) or a solid torus (intersecting ∂M in another annulus), and a proper Möbius strip A ⊂ M is called essential if A does not cut off from M a solid Klein bottle (intersecting ∂M in anotherMöbius strip). The definitions of essential annulus and Möbius strip are moregeneral, but in a P -irreducible and boundary-irreducible M these definitionsare equivalent to the general ones, and for the purpose of this paper we do notneed the general ones.In what follows, we will need to remove some open balls from the manifold M such that their closures and the components of the boundary of M are pairwisedisjoint. In such a case, the manifold obtained will be denoted by ˙ M . Note thatthe boundary of ˙ M is made up of ∂M and some spheres. We allow the casewhere some components of ∂M are spheres, and in this case they are regardedto be different with respect to the spheres in ∂ ˙ M \ ∂M . If instead M is closed,the boundary of ˙ M is made up of the boundary of the removed balls only. Witha slight abuse of notation, we will call ˙ M to be a punctured M . Dehn surfaces
A subset Σ of M is said to be a Dehn surface of M [18] if thereexists an abstract (possibly non-connected) closed surface S and a transverseimmersion f : S → M such that Σ = f ( S ) .Let us fix for a while f : S → M a transverse immersion (hence, Σ = f ( S ) is a Dehn surface of M ). By transversality, the number of pre-images of a pointof Σ is 1, 2 or 3; so there are three types of points in Σ , depending on thisnumber; they are called simple , double or triple , respectively. Note that thedefinition of the type of a point does not depend on the particular transverseimmersion f : S → M we have chosen. In fact, the type of a point can be alsodefined by looking at a regular neighbourhood (in M ) of the point, as shown inFig. 1. The set of triple points is denoted by T (Σ) ; non-simple points are called singular and their set is denoted by S (Σ) . From now on, in all figures, triplepoints are always marked by thick dots and the singular set is also drawn thick. (Quasi-)filling Dehn surfaces A Dehn surface Σ of M will be called quasi-filling if a punctured M , ˙ M , collapses to Σ . Moreover, a quasi-filling Σ is called4 M ˙ M of removed balls S S S × [ − ,
1] 2 S B S × [ − ,
1] 1 RP RP RP ×∼ [ − ,
1] 1 any I -bundle over Σ I -bundle over Σ 0
Table 1: The cases where Σ is a surface. filling if its singularities induce a cell-decomposition of Σ ; more precisely,• T (Σ) = ∅ ,• S (Σ) \ T (Σ) is made up of intervals (called edges ),• Σ \ S (Σ) is made up of discs (called regions ).Since M is connected, any quasi-filling Dehn surface Σ of M is connected.By construction, the complement of Σ in M is made up of components of twotypes:• the components intersecting ∂M , which are all together isomorphic to ∂M × [0 , , with ∂M corresponding to ∂M × { } ;• the components intersecting ∂ ˙ M \ ∂M , which are open balls.Moreover, note that Σ is also a quasi-filling Dehn surface of ˙ M , with all com-ponents of ˙ M \ Σ being of the former type. Finally, note that any small regularneighbourhood R (Σ) of Σ in M is isomorphic to ˙ M . Remark 1. If M is closed, a Dehn surface Σ of M is quasi-filling if M \ Σ ismade up of balls. Therefore, this notion generalises the notion of (quasi-)fillingDehn surfaces of closed orientable 3-manifolds to compact ones [17, 2]. Remark 2.
Suppose a quasi-filling Σ of M is a surface (i.e. S (Σ) = ∅ ). Then, R (Σ) ∼ = ˙ M is an I -bundle over Σ . Since the boundary of ˙ M is the union of ∂M and spheres, and since Σ is connected, we have four cases as described inTable 1.Let us give some other examples. Two projective planes intersecting along aloop non-trivial in both of them, which will be called double projective plane anddenoted by × RP , form a quasi-filling Dehn surface (without triple points) of RP or RP with up to 2 balls removed. A sphere intersecting a torus (resp. aKlein bottle) along a loop is a quasi-filling Dehn surface (without triple points)of S × S (resp. S ×∼ S ) or S × S (resp. S ×∼ S ) with up to 2 balls removed.A sphere self-intersecting in a loop is a quasi-filling Dehn surface (without triplepoints) of a solid torus or of a solid Klein bottle (depending on the orientation ofthe self-intersection) with up to 2 balls removed. The quadruple hat (i.e. a discwhose boundary is glued four times along a circle) is a quasi-filling Dehn-surface(without triple points) of the lens-space L , or of L , with 1 ball removed. Ifwe identify the sphere S with R ∪ {∞} , the three coordinate planes in R ,with ∞ added, form a filling Dehn surface (with two triple points: (0 , , and ∞ ) of S or S with up to 8 balls removed.5igure 2: A cubulation of the 3-dimensional torus S × S × S with two cubes(the identification of each pair of faces is the obvious one, i.e. the one withouttwists).Figure 3: A cube with small tetrahedra removed near the vertices. Ideal cubulations and duality
Consider the topological space c M obtainedfrom ˙ M by collapsing each boundary component (of ˙ M ) to a point. Note thatthe complement of the points corresponding to the boundary components of ˙ M can be identified with the interior of ˙ M , and that c M can be obtained also from M by collapsing each boundary component (of M ) to a point.An ideal cubulation C of M is a cell-decomposition of c M such that• the set of 0-cells (called vertices ) is the set of the points corresponding tothe boundary components of ˙ M ,• each 2-cell (called a face ) is glued along 4 edges,• each 3-cell (called a cube ) is glued along 6 faces arranged like the boundaryof a cube.Note that self-adjacencies and multiple adjacencies are allowed. In Fig. 2 wehave shown a cubulation of the 3-dimensional torus S × S × S with two cubes(a closed manifold). There are two types of vertices in C : those correspondingto the boundary components of M and those corresponding to the spheres of ∂ ˙ M \ ∂M . We will call ideal the former ones and finite the latter ones. Notethat the whole ˙ M can be obtained from C by removing small tetrahedra in eachcube near the vertices (see Fig. 3), that the complement of the ideal verticescan be identified with the interior of M , and that M can be obtained from C by removing small tetrahedra in each cube near the ideal vertices. It is worthnoting that C does not distinguish the ideal vertices corresponding to spheres6igure 4: Local behaviour of duality.from the finite ones. Note also that ours is a slight abuse of notation, indeed inthe classical meaning an ideal polyhedron has only ideal vertices.The following construction is well-known for cubulations (see [1, 9, 5], forinstance), but applies also to ideal cubulations. Let C be an ideal cubulationof M . Consider, for each cube of C , the three squares shown in Fig. 4. Wecan suppose that the squares fit together through the faces, so the subset of M obtained by gluing together these squares is a filling Dehn surface Σ of M .Conversely, an ideal cubulation C of M can be constructed from a filling Dehnsurface Σ of M by considering an abstract cube for each triple point of Σ andby gluing the cubes together along the faces, with the identification of each pairof faces chosen by following the four germs of regions adjacent to the respectiveedge of Σ . The ideal cubulation and the filling Dehn surface constructed in sucha way are said to be dual to each other. Existence of filling Dehn surfaces and reconstruction of the manifold
It is by now well-known that every closed M has a filling Dehn-surface (see,for instance, [17, 21, 2] and also [8] for the quasi-filling case). We will adaptthe proof of [2] to the non-closed case. In what follows, as we have done abovefor cubulations, with a slight abuse of notation, we use a non classical notionof ideal triangulation , i.e. a triangulation T of c M whose vertices are the pointscorresponding to the boundary components of ˙ M . We have ideal and finite vertices: the former being those corresponding to the boundary components of M , the latter being those corresponding to the spheres of ∂ ˙ M \ ∂M . Theorem 3. • Each (connected and compact) 3-manifold has a filling Dehnsurface. • If Σ and Σ are homeomorphic filling Dehn surfaces of (connected andcompact) 3-manifolds M and M , respectively, with the same number ofspherical boundary components, then M and M are also homeomorphic.Proof. We start by proving the first point. Let T be an ideal triangulation ofa 3-manifold M (all 3-manifolds have ideal triangulations, as shown in [16], forinstance). Consider, for each tetrahedron of T , 4 triangles close and parallel tothe 4 faces, as shown in Fig. 5. We can suppose that the triangles fit togetherthrough the faces, so the subset of M obtained by gluing together these trianglesis a Dehn surface Σ of M . It is very easy to prove that Σ is filling, so we leaveit to the reader. 7igure 5: Construction of a filling Dehn sphere from an ideal triangulation.We do not give a complete proof of the second point because it is essentiallythe same as that of Casler for standard spines [6]. The idea of the proof is thefollowing. Let C i be the cubulation of M i dual to Σ i , for i = 1 , . The cubula-tions C i are defined unambiguously, because the cubes dual to the triple pointsand the face identifications of them are defined unambiguously (up to home-omorphism) from the Dehn surfaces Σ i . Since Σ and Σ are homeomorphic,the cubulations C and C turn out to be isomorphic and hence ˙ M and ˙ M arehomeomorphic. The fact that M and M have the same number of sphericalboundary components easily implies that they are obtained from ˙ M and ˙ M ,respectively, by filling up the same number of boundary spherical components,so they are homeomorphic.With respect to the construction described in the first part of the proofabove, it is worth noting that it is the dual counterpart of the well-knownconstruction consisting in dividing a tetrahedron into 4 cubes [20, 7, 9]. Notealso that the surface S such that Σ = f ( S ) is ∂ ˙ M , because Σ can be obtainedby starting with a copy of ∂ ˙ M parallel to ∂ ˙ M in the interior of ˙ M and thenmoving it away from the boundary. Finally, note that Σ has 4 triple points foreach tetrahedron. Remark 4.
With respect to the construction described in the second part ofthe proof above, if we require that the M i ’s are P -irreducible and different fromthe ball B , the M i ’s have no boundary spherical component, so we must fill upall spherical boundary components of the ˙ M i ’s to recover the M i ’s, and hencethe M i ’s are homeomorphic. Abstract filling Dehn surfaces
A filling Dehn surface Σ of M is containedin M . However, one can think of it as an abstract cell complex. For the sake ofcompleteness, we mention that the abstract cell complex Σ determines ˙ M and M if the number of spherical boundary components is known (and the abstractsurface S such that Σ = f ( S ) where f : S → M ) up to homeomorphism (theproof is the same as that of the second part of Theorem 3). This is not truefor quasi-filling Dehn surfaces, e.g. a surface Σ is a quasi-filling Dehn surface ofany I -bundle over Σ . Surface-complexity
The surface-complexity of M can now be defined as theminimal number of triple points of a quasi-filling Dehn surface of M . More8recisely, we give the following. Definition 5.
The surface-complexity sc ( M ) of M is equal to c if M possessesa quasi-filling Dehn surface with c triple points and has no quasi-filling Dehnsurface with less than c triple points. In other words, sc ( M ) is the minimum of | T (Σ) | over all quasi-filling Dehn surfaces Σ of M .We will classify the 3-manifolds having surface-complexity zero in the follow-ing section. At the moment, we can only say that S , B , RP , S × S , S ×∼ S , L , , the I -bundles and some of these manifolds with a certain number of ballsremoved have surface-complexity 0, because we have seen above that they havequasi-filling Dehn surfaces without triple points. A quasi-filling Dehn surface Σ of M is called minimal if it has a minimal numberof triple points among all quasi-filling Dehn surfaces of M , i.e. | T (Σ) | = sc ( M ) . Theorem 6.
Let M be a (connected and compact) P -irreducible and boundary-irreducible 3-manifold without essential annuli and Möbius strips. • If sc ( M ) = 0 , the manifold M is the sphere S , the ball B , the projectivespace RP or the lens space L , . • If sc ( M ) > , the manifold M has a minimal filling Dehn surface.Proof. Let Σ be a minimal quasi-filling Dehn surface of M . If we have S (Σ) = ∅ (i.e. Σ is a surface), by virtue of Remark 2, we have that M is S , B , RP or an I -bundle. The last case cannot occur, because an I -bundle is not P -irreducible,or contains essential annuli or Möbius strips.Then, we suppose S (Σ) = ∅ . We will first prove that M has a quasi-fillingDehn surface Σ ′ such that Σ ′ \ S (Σ ′ ) is made up of discs or Σ ′ is a surface.Suppose there exists a component C of Σ \ S (Σ) that is not a disc. C containsa non-trivial orientation preserving (in C ) simple closed curve γ . Consider astrip F properly embedded in ˙ M such that F ∩ Σ = γ . The strip F is eitheran annulus or a Möbius strip depending on whether γ is orientation preservingin M or not. Since M \ ˙ M is made up of balls, we can fill up ∂F with two,one or zero discs in M \ ˙ M , depending on whether the number of componentsof F ∩ ( ∂ ˙ M \ ∂M ) are two, one or zero, respectively. We then get a surface F properly embedded in M . If F is an annulus, F is a sphere, a disc or F itself.If F is an Möbius strip, F is a projective plane or F itself. Therefore, we havefive cases to analyse.• If F is a sphere, it bounds a ball (in M ), say B , because M is P -irreducible. Since Σ ∩ ∂B = γ is a simple closed curve, we can replace theportion of Σ contained in B with a disc, getting a new quasi-filling Dehnsurface of M .• If F is a disc, it cuts off a ball from M (intersecting ∂M in anotherdisc), say B , because M is P -irreducible and boundary-irreducible. Since Σ ∩ ∂B = γ is a simple closed curve, we can replace the portion of Σ contained in B with a disc, getting a new quasi-filling Dehn surface of M .9 If F is the annulus F , it cuts off from M either a cylinder (intersecting ∂M in two discs), say B , or a solid torus (intersecting ∂M in anotherannulus), say T , because M is P -irreducible, boundary-irreducible andwithout essential annuli. In the former case, since Σ ∩ ∂B = γ is a simpleclosed curve, we can replace the portion of Σ contained in B with a disc,getting a new quasi-filling Dehn surface of M . The latter case insteadcannot occur, because the core of F would be a longitude of the solidtorus T and would be null-homologous in T (indeed Σ ∩ T is a 2-cyclewhose boundary is the core of F ), a contradiction.• If F is a projective plane, it is two-sided (because F is), which is notpossible because M is P -irreducible.• If F is the Möbius strip F , it cuts off from M a solid Klein bottle (inter-secting ∂M in another Möbius strip), say K , because M is P -irreducible,boundary-irreducible and without essential Möbius strips. This case can-not occur, because the core of F would be a longitude of the Klein bottle K and would be null-homologous in K (indeed Σ ∩ K is a 2-cycle whoseboundary is the core of F ), a contradiction.In all admissible cases the Euler characteristic of the component of Σ \ S (Σ) containing γ has increased, no new non-disc component has been created andthe number of triple points is not increased. Hence, by repeatedly applying thisprocedure, we eventually get a quasi-filling Dehn surface, say Σ ′ , of M such that Σ ′ \ S (Σ ′ ) is made up of discs or Σ ′ is a surface.If Σ ′ is a surface, analogously to what said at the beginning of the proof for Σ , by virtue of Remark 2, we have that M is S , B or RP .Therefore, we eventually consider the case where Σ ′ \ S (Σ ′ ) is made up ofdiscs. Since Σ ′ is connected, also S (Σ ′ ) is connected. If we have sc ( M ) > (i.e. T (Σ ′ ) is not empty), S (Σ ′ ) \ T (Σ ′ ) cannot contain circles and hence Σ ′ is filling (i.e. M has a minimal filling Dehn surface). Otherwise, if we have sc ( M ) = 0 (i.e. T (Σ ′ ) is empty), S (Σ ′ ) is made up of one circle. Since Σ ′ \ S (Σ ′ ) is made up of discs, the Dehn surface Σ ′ is completely determined by the regularneighbourhood of S (Σ ′ ) in Σ ′ . This neighbourhood depends on how the germsof disc are interchanged along the curve S (Σ ′ ) . Since M is P -irreducible, onlythree cases must be taken into account for Σ ′ (up to symmetry):• two spheres intersecting along the circle S (Σ ′ ) , which form a Dehn surfaceof S or B ;• the double projective plane × RP , which is a Dehn surface of RP ;• the four-hat, which is a Dehn surface of L , .This concludes the proof.Since there is a finite number of filling Dehn surfaces having a fixed number oftriple points and each of them is the Dehn surface of one P -irreducible manifold(two for the S / B case – see Remark 4), and since there are only a finite numberof P -irreducible and boundary-irreducible 3-manifolds without essential annuliand Möbius strips with surface-complexity 0, we have the following result.10igure 6: Bubble-move. Corollary 7.
For any integer c there exists only a finite number of (connectedand compact) P -irreducible and boundary-irreducible 3-manifolds without es-sential annuli and Möbius strips that have surface-complexity c . By means of the duality between filling Dehn surfaces and ideal cubulationswe have the following result.
Corollary 8.
The surface-complexity of a (connected and compact) P -irreducibleand boundary-irreducible 3-manifold without essential annuli and Möbius strips,different from S , B , RP and L , , is equal to the minimal number of cubes inan ideal cubulation of M . Theorem 6 states that, under some hypotheses, M has a minimal filling Dehnsurface, but not all minimal quasi-filling Dehn surfaces of such an M are indeedfilling. However, we will see now that they can be all constructed starting fromfilling ones and applying a simple move. The move acts on quasi-filling Dehnsurfaces near a simple point as shown in Fig. 6 and it is called a bubble-move .Note that the result of applying a bubble-move to a quasi-filling Dehn surface of M is a quasi-filling Dehn surface of M , but the result of applying a bubble-moveto a filling Dehn-surface is not filling any more. Note also that the bubble-moveincreases (by two) the number of balls in M \ ˙ M . If a quasi-filling Dehn surface Σ is obtained from a quasi-filling Dehn surface Σ by repeatedly applying bubble-moves (even zero), we will say that Σ is derived from Σ . Obviously, if Σ is aquasi-filling Dehn surface of M and is derived from Σ , also Σ is a quasi-fillingDehn surface of M . Eventually, it is worth noting (even if not useful for thepurpose of this paper) that if we have in Σ a configuration as in Fig. 6-right, wecan apply an inverse bubble move, removing the sphere from Σ , only if the ballbounded by the sphere in Fig. 6-right is actually a ball in M , i.e. it containstwo balls of M \ ˙ M .We will need the following result, which has been proved in [3, Lemma 2.3]. Lemma 9.
Let Σ be a minimal quasi-filling Dehn surface of the sphere S andlet D be a closed disc contained in Σ \ S (Σ) . Then Σ is derived from a sphere S by means of bubble-moves not involving D . Theorem 6 can be improved by means of a slightly subtler analysis.
Theorem 10.
Let Σ be a minimal quasi-filling Dehn surface of a (connectedand compact) P -irreducible and boundary-irreducible 3-manifold M without es-sential annuli and Möbius strips. • If sc ( M ) = 0 , one of the following holds: – M is the sphere S or the ball B , and Σ is derived from the sphere S ; M is the projective space RP , and Σ is derived from the projectiveplane RP or from the double projective plane × RP ; – M is the lens space L , , and Σ is derived from the four-hat. • If sc ( M ) > , the Dehn surface Σ is derived from a minimal filling Dehnsurface of M .Proof. The scheme of the proof is the same as that of Theorem 6. Hence, wewill often refer to the proof of Theorem 6 also for notation.Let Σ be a minimal quasi-filling Dehn surface of M . If we have S (Σ) = ∅ (i.e. Σ is a surface), by virtue of Remark 2, we have three cases:• Σ is S and M is S or B ;• Σ is RP and M is RP ;• Σ is a surface and M is an I -bundle, which is not P -irreducible, or containsessential annuli or Möbius strips, so this case cannot occur.Then, we suppose S (Σ) = ∅ . We will first prove that Σ is derived froma (minimal) quasi-filling Dehn surface Σ ′ of M such that either Σ ′ \ S (Σ ′ ) ismade up of discs or Σ ′ is a surface. Suppose there exists a component C of Σ \ S (Σ) that is not a disc. C contains a non-trivial orientation preserving (in C ) simple closed curve γ . As done in the proof of Theorem 6, we consider astrip F properly embedded in ˙ M such that F ∩ Σ = γ , and fill up ∂F withtwo, one or zero discs in M \ ˙ M , getting a surface F properly embedded in M .Therefore, we have five cases to analyse.• If F is a sphere, it bounds a ball (in M ), say B , because M is P -irreducible. Consider Σ B = Σ ∩ B and Σ M = Σ \ B . If we fill up Σ M witha disc by gluing it along γ , we obtain a minimal quasi-filling Dehn surface Σ ′ M of M . Analogously, if we fill up Σ B with a disc (say D ) by gluingit along γ , we obtain a minimal quasi-filling Dehn surface Σ ′ B of S . Byvirtue of Lemma 9, Σ ′ B is derived from a sphere S by means of bubble-moves not involving D . These moves can be applied to Σ ′ M because theydo not involve D and the result is Σ , so Σ is derived from the quasi-fillingDehn surface Σ ′ M of M .• If F is a disc, it cuts off a ball from M (intersecting ∂M in another disc),say B , because M is P -irreducible and boundary-irreducible. Consider Σ B = Σ ∩ B and Σ M = Σ \ B . If we fill up Σ M with a disc by gluing it along γ , we obtain a minimal quasi-filling Dehn surface Σ ′ M of M . Analogously,if we fill up Σ B with a disc (say D ) by gluing it along γ , we obtain aminimal quasi-filling Dehn surface Σ ′ B of S . By virtue of Lemma 9, Σ ′ B is derived from a sphere S by means of bubble-moves not involving D .These moves can be applied to Σ ′ M because they do not involve D and theresult is Σ , so Σ is derived from the quasi-filling Dehn surface Σ ′ M of M .• If F is the annulus F , it cuts off from M either a solid torus (intersect-ing ∂M in another annulus), say T , or a cylinder (intersecting ∂M intwo discs), say B , because M is P -irreducible, boundary-irreducible andwithout essential annuli. The former case cannot occur (see the proof ofTheorem 6). The latter case instead can occur. Consider Σ B = Σ ∩ B Σ M = Σ \ B . If we fill up Σ M with a disc by gluing it along γ , weobtain a minimal quasi-filling Dehn surface Σ ′ M of M . Analogously, if wefill up Σ B with a disc (say D ) by gluing it along γ , we obtain a minimalquasi-filling Dehn surface Σ ′ B of S . By virtue of Lemma 9, Σ ′ B is derivedfrom a sphere S by means of bubble-moves not involving D . These movescan be applied to Σ ′ M because they do not involve D and the result is Σ ,so Σ is derived from the quasi-filling Dehn surface Σ ′ M of M .• The last two cases, where F is a projective plane or the Möbius strip F , cannot occur because M is P -irreducible, boundary-irreducible andwithout essential Möbius strips (see the proof of Theorem 6).In all admissible cases the Euler characteristic of the component of Σ ′ M \ S (Σ ′ M ) containing γ is bigger than that of the corresponding component of Σ \ S (Σ) ,no new non-disc component has been created and the number of triple pointshas not changed. Hence, by repeatedly applying this procedure, we eventuallyget a (minimal) quasi-filling Dehn surface Σ ′ of M from which Σ is derived andsuch that either Σ ′ \ S (Σ ′ ) is made up of discs or Σ ′ is a surface.If Σ ′ is a surface, analogously to what said at the beginning of the proof for Σ , by virtue of Remark 2, we have two cases:• Σ ′ is S and M is S or B ;• Σ ′ is RP and M is RP .Therefore, we eventually consider the case where Σ ′ \ S (Σ ′ ) is made up ofdiscs. Since Σ ′ is connected, also S (Σ ′ ) is connected. If we have sc ( M ) > (i.e. T (Σ ′ ) is not empty), S (Σ ′ ) \ T (Σ ′ ) cannot contain circles and hence Σ ′ isfilling (i.e. Σ is derived from a minimal filling Dehn surface of M ). Otherwise, ifwe have sc ( M ) = 0 (i.e. T (Σ ′ ) is empty), S (Σ ′ ) is made up of one circle. Since Σ ′ \ S (Σ ′ ) is made up of discs, the Dehn surface Σ ′ is completely determinedby the regular neighbourhood of S (Σ ′ ) in Σ ′ . This neighbourhood dependson how the germs of disc are interchanged along the curve S (Σ ′ ) . Since M is P -irreducible, only three cases must be taken into account for Σ ′ (up tosymmetry):• two spheres intersecting along the circle S (Σ ′ ) , which form a Dehn surfaceof S or B ;• the double projective plane × RP , which is a Dehn surface of RP ;• the four-hat, which is a Dehn surface of L , .We conclude the proof by noting that in the first case Σ ′ is derived from thesphere S . An important feature of a complexity function is to behave well with respectto the cut-and-paste operations. In this section, we will prove that the surface-complexity is subadditive under (boundary) connected sum.13 Σ Σ B (cid:1) D Figure 7: The Dehn surface Σ ∪ Σ in ˙ M ∂ ˙ M with the arc α (left), itsmodification Σ (centre), and the ball B to be removed to get another punctured M ∂ M (right). Theorem 11.
The surface-complexity of the connected sum and of the boundaryconnected sum of (connected and compact) 3-manifolds is less than or equal tothe sum of their surface-complexities.Proof.
In order to prove the theorem, it is enough to prove the statementin the case where the number of the manifolds involved in the (boundary)connected sum is two. Hence, if we call M and M the two manifolds, weneed to prove that sc ( M M ) sc ( M ) + sc ( M ) and that sc ( M ∂ M ) sc ( M ) + sc ( M ) . Let Σ (resp. Σ ) be a quasi-filling Dehn surface of M (resp. M ) with sc ( M ) (resp. sc ( M ) ) triple points, and let ˙ M (resp. ˙ M ) bethe punctured M (resp. M ).We start from the boundary connected sum. It is obtained by identifying twodiscs D and D in ∂M and ∂M , respectively. Call D the corresponding disc in M ∂ M , which is properly embedded. Since the discs D and D are containedalso in ∂ ˙ M and ∂ ˙ M , respectively, we can consider also the (restriction ofthe) boundary connected sum ˙ M ∂ ˙ M , which turns out to be a punctured M ∂ M . We can suppose that Σ and Σ are embedded also in ˙ M ∂ ˙ M .Consider an embedded arc α connecting Σ and Σ in ˙ M ∂ ˙ M as shown inFig. 7-left. The boundary connected sum ˙ M ∂ ˙ M collapses to Σ ∪ Σ ∪ α .The last object is not a Dehn sphere (because it has a 1-dimensional part), butwe can modify Σ ∪ Σ as shown in Fig. 7-centre, getting a Dehn surface, say Σ .If we remove a small ball B (see Fig. 7-right) from ˙ M ∂ ˙ M we obtain anotherpunctured M ∂ M which collapses to Σ . Therefore, Σ is a quasi-filling Dehnsurface of M ∂ M with sc ( M ) + sc ( M ) triple points and hence we have sc ( M ∂ M ) sc ( M ) + sc ( M ) .For the connected sum we note that if we remove balls B and B from M and M , respectively, and we apply a boundary connected sum along discscontained in ∂B and ∂B , we get M M minus a ball (whose boundary isthe union of the complement of the discs in ∂B and ∂B ). For i = 1 , , up toapplying a bubble-move (which does not increase the number of triple points),we can suppose that ˙ M i is obtained from M i by removing at least one ball, andwe can use one of the balls in M i \ ˙ M i as the ball B i for the construction of theconnected sum. By applying the construction above to the boundary connectedsum along discs contained in ∂B and ∂B , we get ˙ M ∂ ˙ M and a Dehn surface Σ to which ( ˙ M ∂ ˙ M ) \ B collapses (where B is the small ball of Fig. 7-right).Since ( ˙ M ∂ ˙ M ) \ B is a punctured ( M \ B ) ∂ ( M \ B ) and the latter is M M minus one ball, ( ˙ M ∂ ˙ M ) \ B is also a punctured M M . Therefore, Σ is a quasi-filling Dehn surface of M M with sc ( M ) + sc ( M ) triple points14igure 8: Construction of an ideal triangulation from an ideal cubulation.Figure 9: Inserting a tetrahedron between two pairs of triangles not matchingeach other.and hence we have sc ( M M ) sc ( M ) + sc ( M ) . In general, calculating the surface-complexity sc ( M ) of M is very difficult, how-ever it is relatively easy to estimate it. More precisely, it is quite easy to giveupper bounds for it. Obviously, if we construct a quasi-filling Dehn surface Σ of M , the number of triple points of Σ is an upper bound for the surface-complexityof M .The construction described in the first part of Theorem 3, obviously impliesthe following result. Theorem 12.
If a (connected and compact) 3-manifold M has an ideal trian-gulation with n tetrahedra, the inequality sc ( M ) n holds. From ideal cubulations to ideal triangulations
We will now describe theinverse construction, allowing us to create ideal triangulations from filling Dehnsurfaces (or ideal cubulations, by duality). Let C be an ideal cubulation of M .Consider, for each cube of C , the five tetrahedra shown in Fig. 8. The idea isto glue together these “bricks” (each of which is made up of 5 tetrahedra) byfollowing the identifications of the faces of C . Note that each face of the cubes isdivided by a diagonal into two triangles and that it may occur that these pairsof triangles do not match each other. If this occurs, we insert a tetrahedronbetween them as shown in Fig. 9. Eventually, we get an ideal triangulation T of M with five tetrahedra for each cube of C and at most one tetrahedron foreach face of C . Since the number of faces of an ideal cubulation is three times15he number of cubes, we have that the number of tetrahedra in T is at most 8times the number of cubes.We note that there are two different identifications of the abstract “brick”with each cube, so if there are k cubes there are k possibilities for the identifi-cations with the cubes of C . Some of them may need less insertions of tetrahedra(for matching the pairs of triangles in the faces of C ) than others. Hence, op-timal choices may lead to a triangulation of M with a number of tetrahedracloser to 5 times the number of cubes. Matveev complexity
The Matveev complexity [15] of a (connected and com-pact) 3-manifold M is defined using simple spines. A polyhedron P is simple if the link of each point of P can be embedded in the 1-skeleton of the tetra-hedron. The points of P whose link is the whole 1-skeleton of the tetrahedronare called vertices . If ∂M = ∅ , a sub-polyhedron P of M is a spine of M if M collapses to P . If ∂M = ∅ , a sub-polyhedron P of M is a spine of M if M minus a ball collapses to P . The Matveev complexity c ( M ) of M is the minimalnumber of vertices of a simple spine of M . If M is P -irreducible, boundary-irreducible, without essential annuli and Möbius strips, and different from S , B , RP and L , , its Matveev complexity is the minimal number of tetrahedrain an ideal triangulation of its (the Matveev complexity of S , B , RP and L , is ), see [16, 14]. Therefore, the Matveev complexity is related to the surface-complexity, indeed the constructions described in Theorem 3 and above, and thelist of the P -irreducible and boundary-irreducible 3-manifolds without essen-tial annuli and Möbius strips with Matveev complexy zero or surface-complexityzero obviously imply the following. Theorem 13.
Let M be a (connected and compact) P -irreducible and boundary-irreducible 3-manifold without essential annuli and Möbius strips, different fromthe lens spaces L , and L , ; then the inequalities sc ( M ) c ( M ) and c ( M ) sc ( M ) hold. Moreover, we have c ( L , ) = 0 , sc ( L , ) > , c ( L , ) > and sc ( L , ) = 0 . A The bidimensional case
Let S be a connected and compact surface. Let us denote by ( S ) ⊔ n the disjointunion of n circles S . A subset Γ of S is said to be a Dehn loop of S if there exists n ∈ N and a transverse immersion f : ( S ) ⊔ n → S such that Γ = f (( S ) ⊔ n ) .Some examples are shown in Fig. 10. There are only two types of points in Γ :smooth points and crossing points (see Fig. 11). The set of crossing points isdenoted by C (Γ) . As for 3-manifolds, we will need to remove some open discsfrom the surface S such that their closures and the components of the boundaryof S are pairwise disjoint; we will denote the surface obtained by ˙ S and call it a punctured S . A Dehn loop Γ of S will be called quasi-filling if a punctured S , ˙ S , collapses to Γ . A quasi-filling Γ will be called filling if its singularities inducea cell-decomposition of Γ : more precisely,• C (Γ) = ∅ ,• Γ \ C (Γ) is made up of intervals.16igure 10: Some Dehn loops of the sphere S , the projective plane RP , thetorus minus a disc T \ D and the Klein bottle K .Figure 11: Smooth points (left) and crossing points (right).Note that the latter condition is automatically satisfied once the former is ful-filled (because we are taking into account only connected surfaces).Note that the Euler characteristic of ˙ S is equal to the Euler characteristic ofthe Dehn loop Γ . On the other hand, if we start from an abstract Dehn loop Γ and we thicken it to produce a surface with boundary, we can obtain differentsurfaces with boundary with the same Euler characteristic of Γ by choosing the“twists” along the loops of Γ . A simple case-by-case check implies that eachsurface S has a filling Dehn loop. Moreover, only the sphere S , the ball B ,the annulus A , the projective plane RP and the Möbius strip M have quasi-filling Dehn loops that are not filling, while all other surfaces have only fillingDehn loops. However, as opposed to the 3-dimensional case (Theorem 3), afilling Dehn loop does not determine S (even knowing the number of boundarycomponents of ˙ S that must be filled up); for instance, the bouquet of two circlesis a Dehn loop of the projective plane RP , the projective plane RP with up totwo discs removed, the torus T , the torus T with one disc removed, the Kleinbottle K and the Klein bottle K with one disc removed (see Fig. 10).We define the loop-complexity lc ( S ) of a connected and compact surface S as the minimal number of crossing points of a quasi-filling Dehn loop of S .For a non-closed S different from B , A and M , by means of an easy Eulercharacteristic argument (in dimension 2 it is a very powerful tool) and a case-by-case analysis, it is quite easy to prove that the minimal number of crossing pointsof a filling Dehn surface of S occurs when there is no puncture (i.e. S = ˙ S ) andthat this number of crossing points is the opposite of the Euler characteristicof S . Since for closed surfaces a puncture is needed (to create the boundaryfor collapsing), we obtain that the loop-complexity of a surface S with Eulercharacteristic χ is• − χ if ∂S = ∅ , except for B having loop-complexity 0;• − χ if ∂S = ∅ , except for S having loop-complexity 0.Note that it is not true that the loop-complexity of the connected sum of twosurfaces is at most the sum of their loop-complexities; for instance, we have17 = RP RP while lc ( K ) = 1 lc ( RP ) + lc ( RP ) .We can also consider, as done for 3-manifolds, ideal cubulations C of con-nected and compact surfaces S , i.e. cell-decompositions of the topological space b S obtained from ˙ S by collapsing each boundary component (of ˙ S ) to a pointsuch that each 2-cell (a square ) is glued along four edges. An ideal cubulation of S can be constructed from a filling Dehn loop Γ of S by considering an abstractsquare for each crossing point of Γ and by gluing the squares together alongthe edges. However, there are two possibilities for gluing two squares along anedge and the abstract polyhedron Γ does not encode any information to choosethe right one. (In some sense, this explains why a filling Dehn loop does notdetermine unambiguously one surface.) If we consider also the immersion of ( S ) ⊔ n in the surface S containing the Dehn loop Γ , we can choose the rightidentifications and construct an ideal cubulation C of S .The converse construction can also be performed, obtaining a filling Dehnloop of a connected and compact surface S from an ideal cubulation of S . Theseconstructions tell us that lc ( S ) is the minimal number of squares in a cubulationof S , except for S , B , A , RP and M , whose loop-complexity is . References [1]
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