Heegaard splittings of graph manifolds
Enrique Artal Bartolo, Simón Isaza Peñaloza, Miguel Marco-BuzunÁriz
aa r X i v : . [ m a t h . G T ] F e b HEEGAARD SPLITTINGS OF GRAPH MANIFOLDS.
ENRIQUE ARTAL BARTOLO, SIM ´ON ISAZA PE ˜NALOZA,AND MIGUEL A. MARCO-BUZUN ´ARIZ
Abstract.
In this paper we give a method to construct Heegaard splittings oforiented graph manifolds with orientable bases. A graph manifold is a closed 3-manifold admitting only Seifert-fibered pieces in its Jaco-Shalen decomposition;for technical reasons, we restrict our attention to the fully oriented case, i.e.both the pieces and the bases are oriented.
In this paper we deal with graph manifolds. A closed 3-manifold M is saidto be a graph manifolds if its Jaco-Shalen decomposition admits only Seifert-fibered pieces. These manifolds were classified by F. Waldhausen [14, 15] andthey are completely determined by a normalized weighted graph (up to a controlledfamily of exceptions). For technical reasons we restrict our attention to the fully oriented case, i.e. we assume M oriented and we also assume that the bases ofthe Seifert fibrations are oriented surfaces. This is only a mild restriction andthis family contains the class of 3-manifolds which appear naturally in complexgeometry: boundary of regular neighbourhoods of complex curves in complexsurfaces, and, in particular links of normal surface complex singularities. Thesemanifolds admit another classification in terms of plumbing graphs, see the workof W. Neumann [8].A Heegaard splitting of a closed orientable 3-manifold M is a decompositionof M as a union of two handle bodies sharing a common boundary. This commonboundary is a closed orientable surface Σ. The genus of the splitting is defined asthe genus g of Σ. Note that, if we see Σ as the boundary of a handle body, thereare g distinguished curves in it, that correspond to the boundaries of g disks suchthat, cutting along them, a closed ball is obtained. In a Heegaard splitting, thesame surface is seen as the boundary of two different handle bodies, so there aretwo families of distinguished curves. These two families of curves are enough todetermine the two handle bodies, and hence they also determine the manifold M and the splitting itself. An oriented closed surface of genus g , with two families of g curves is called a Heegaard diagram , which represents a Heegaard splitting. Everyclosed oriented 3-manifold admits a Heegaard splitting [4], and [11] for details.The
Heegaard genus of such a manifold is the minimal genus of the Heegaardsplittings of M . There are a lot of works about Heegaard splittings of Seifert fibered manifolds(the bricks of graph manifolds), see e.g. [3, 2, 6]. In these works, vertical and hor-izontal splittings are defined; our approach will make use of horizontal splittings.These ideas were also transferred to the case of graph manifolds in [13], where thestructure of Heegaard splittings is studied.The contribution of this work is to provide an explicit method to constructHeegaard splittings of a graph manifold from its plumbing graph, namely, we givea closed oriented surface with two systems of cutting curves. Recall from [8] thatsome moves are allowed for plumbing graphs that provide the same manifold; wecan use these moves to decrease the genus of the provided Heegaard splitting eventhough, in general, our method does not provide a minimal splitting.Osv´ath and Szab´o [9, 10] defined a Floer homology for 3-manifolds using Hee-gaard diagrams (the so-called Heegaard-Floer homology). Since then, Heegaardsplittings have regained interest, specially when having combinatorial methods forits computation from a Heegaard diagram, see Sarkar and Wang in [12]. An inter-esting particular case is its application to the study of normal surface singularityinvariants, specially those whose links are rational homology spheres, as in theseries of works of N´emethi et al. [5, 1, 7].The paper is organized as follows. We start in § §
2, we recall theconstruction of a graph manifold from its plumbing graph for further use. In § float gluings . The case of S -fiber bundles withEuler number ± § § S -fiberbundles in §
6. In §
7, we study the splittings of the simplest graph manifolds whichare not fibered bundles, i.e., corresponding to a simplicial graph with one edge.The general case is studied in §
8. This escalonated procedure allows us to splitthe technical difficulties. Finally, in § E singularity).1. illustrative example The goal of this paper is to describe an explicit Heegaard splitting of a graphmanifold. It is presented in the form of a method, that we will now summarizeby describing a surface with two systems of curves starting from of a decoratedgraph. We illustrate this with a suitable example.
EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 3 + − +[1] , − , , v is decorated withtwo numbers: a nonnegative integer [ g v ] and an integer e v . Each edge is decoratedwith a sign.From the graph, we will construct a surface, and two systems of curves inside ofit (refered to as the system of blue curves and the system of red curves), followinga process that mymics the construction of the graph from its elements. In thisprocess we fix a spanning tree that determines two types of edges: edges in the treeand edges that close cycles. In our example we fix as spanning tree the straightedges. The steps to follow are the following:(G1) For each vertex v , we consider a pair of closed oriented surfaces of genus g v (called top and bottom ) as in Figure 1.2 for the example.Figure 1.2(G2) We join the surfaces of each pair by some cylinders, see Figure 1.3. To eachone of these cylinders it will be assigned a sign, satisfying the conditionthat the sum of these signs in each pair of surfaces matches the number e v .The number of these cylinders can be chosen freely, as long as the previouscondition holds, and there are enough of them to perform the rest of thesteps in the algorithm. Besides, one of the cylinders in each pair of surfacesis chosen as a main cylinder (larger in Figure 1.3). E. ARTAL, S. ISAZA, AND M. MARCO − + − + + − +Figure 1.3(G3) For each handle in a surface, see Figure 1.4, we add a red curve that turnsaround the handle meridian, passes to the other surface in the pair throughthe main cylinder, follows the same path in the other surface (reversingdirection) and returns back to the starting point traversing again the maincylinder (without self intersections). Another red curve is constructed inthe same way but following the handle longitudes instead of the meridians.Figure 1.4. Handle red curves of step (G3) for the surfaces of thegenus 1 vertex.(G4) For each cylinder C which is not a main cylinder, we add a red curve thatgoes through the main cylinder and returns through C .(G5) For each red curve, we add a blue curve. These blue curves are parallel tothe red curves, except for performing a Dehn twist around each cylinderthey cross. The direction of the Dehn twist is given by the sign of thecylinder.(G6) Now we add the edges of the graph one by one, starting from the edges inthe tree. To add an edge of sign s in the tree, we choose one cylinder with EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 5
Figure 1.5. Red curves in step (G4).Figure 1.6. All lines added after step (G5).sign s in each of the corresponding pair of surfaces. These cylinders shouldbe crossed only by one blue line (i.e. distinct from the main one, whenthe corresponding surface has either more than two cylinders or positivegenus). Then we substitute these two cylinders with one cylinder thatjoins the upper surfaces, and another one that joins the lower ones. Thered lines are just directly glued. The blue lines are also glued to form anew one. This new blue line goes parallel to the new red line in one ofthe new cylinders, but performs a Dehn twist around the other one. Thedirection of the Dehn twist will be given by the sign s of the edge.(G7) If the edge creates a loop, we choose cylinders and substitute them by newones as before. The two red lines δ r and γ r are substituted by two newones. The first one is constructed as in (G6). In order to construct thesecond one we choose (arbitrarily) one of the old ones, say γ r ; it can bedecomposed as γ · γ where γ is the the path contained in the tube whichis going to disappear. We take two parallel copies of γ and we connect E. ARTAL, S. ISAZA, AND M. MARCO
Figure 1.7. Curves after adding one edge in step (G6).Figure 1.8. Curves after adding the second edge in step (G6)them by turning around the new cylinders in such a way that the resultingcurve is disjoint with the first red curve. Let δ b and γ b be the two old bluelines. As before, a new blue line is obtained by gluing δ b and γ b as in (G6),going parallel to the corresponding red one in one of the new cylinders andperforming a Dehn twist along the other one. The second new blue line iscreated from one of the preexisting ones (say δ b in this example) as we didfor the red one. That is, we decompose δ b as δ · δ where δ is the the pathcontained in the tube which is going to disappear; we take two parallelcopies of δ and we connect them by turning around the new cylinders insuch a way that the resulting curve is disjoint with the first blue curve.2. Plumbing graph of a graph manifold
We recall the needed facts of Neumann’s plumbing construction [8] of Wald-hausen graph manifolds [14, 15]. Everything in this section is known but we recallit in order to fix notations. The atoms of these constructions are S -fiber bundles. EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 7
Figure 1.9. Curves after step (G7). The Handle curves have beenomited for clarity.Since the actual family we are interested in satisfies strong orientation propertieswe will restrict our attention to oriented graph manifolds built up using orientedfibrations.Let π : M → S be an oriented S -bundle over a closed oriented surface S of genus g . The oriented S -bundles over a manifold N are classified by its Eulerclass in H ( N ; Z ). If S is an oriented closed surface there is a natural identification Z ≡ H ( S ; Z ) and the Euler class is interpreted as an Euler number e ∈ Z . Letus recall for further use how to compute this number. Because of the Euler classclassification, any oriented S -bundle over an oriented surface with boundary ishomeomorphic to a product.Let us consider a small closed disk D ⊆ S and consider the surface with bound-ary ˇ S := S \ D . The restrictions of π over D and ˇ S are product bundles. Let µ be the boundary of a meridian disk of the solid torus π − ( D ) (oriented accordinglyas ∂D ) and let s be the boundary of a section defined over ˇ S (oriented as ∂ ˇ S ).These two simple closed curves define elements in H ( π − ( ∂D ); Z ) as an orientedfiber φ does. Let us use multiplicative notation for H ( π − ( ∂D ); Z ). The factthat µ and s project onto opposite generators of H ( ∂D ; Z ) implies that theseelements satisfy a relation(2.1) s · µ · φ e = 1 E. ARTAL, S. ISAZA, AND M. MARCO for some e ∈ Z , which happens to be the Euler number of the fibration. Thereare several variations of this construction. The first one is very simple, we canreplace D, ˇ S by two surfaces S , S with common connected boundary such that S = S ∪ S and the formula (2.1) is still true. Moreover, there is no need to assumethat the their boundaries are connected. Assume that ∂S = ∂S = S ∩ S has r connected components C , . . . , C r ; let us fix sections s i : S i → M , i = 1 ,
2, andlet us denote by s ji the boundary of such section in C j (oriented as ∂S i ). Then in H ( C j ; Z ) we have inequalities(2.2) s j · s j · φ e j = 1 , e j ∈ Z , and e = e + · · · + e r .Moreover, any decomposition of e as above, can be realized in this way for agiven oriented S -bundle with Euler number e .A plumbing graph (Γ , g, e, o ) is given by a (connected) graph Γ (without loops),a genus function g : V (Γ) → Z ≥ (where V (Γ) is the set of vertices of Γ), an Euler function e : V (Γ) → Z and an orientation class o ∈ H (Γ; Z / v with [ g ( v )] and e ( v ), andby decorating each edge e with a sign σ e = ± representing the coefficients of acocycle (cochain) representing o . If the decoration [ g ( v )] is not written it meansthat g ( v ) = 0, and empty decoration of an edge e means +-decoration. Remark . If we change a cocycle by reversing the signs of all the edges adjacentto a fixed vertex, we obtain another representative of o ; moreover, we can passfrom one representative to another by a sequence of these moves. Of course, if Γis a tree the o -decoration can be chosen as void.The plumbing manifold associated to (Γ , g, e, o ) is constructed as follows. First,we collect for each v ∈ V (Γ) an oriented S -bundle π v : M v → S v with Eulernumber e ( v ) and such that S v is a closed oriented surface of genus g ( v ). For eachedge η with end points v, w we collect two closed disks D ηv ⊂ S v and D ηw ⊂ S w .We choose these disks such that they are pairwise disjoint for any fixed v . Let usdefine ˇ M v to be the closure of M v \ S v ∈ η π − v ( D ηv ), which is an oriented manifoldwhose boundary is composed by tori, as many as the valency of v in Γ. We definethen T ηv := π − v ( ∂D ηv ). In each one of these tori we have a pair of curves ( φ ηv , µ ηv ),where µ ηv is a meridian of the solid torus π − v ( D ηv ) (oriented as ∂D ηv ) and φ ηv is anoriented fiber. Note that these curves induce a basis of H ( T ηv ; Z ) which representsthe orientation of T ηv as part of the boundary of ˇ M v .Let us consider a homeomorphism Φ ηv,w : T ηv → T ηw such that Φ ηv,w ( φ ηv ) = ( µ ηw ) σ η and Φ ηv,w ( µ ηv ) = ( φ ηw ) σ η . Basically, we are exchanging sections and fibers (twisted EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 9 by the sign). This map is determined up to isotopy by the matrix σ η ( ) ofdeterminant −
1. These maps are well-defined only up to isotopy and we canchoose representatives such that Φ ηw,v = (Φ ηv,w ) − . Then the plumbing manifoldassociated to (Γ , g, e, o ) is defined as: a v ∈ V (Γ) ˇ M v . { Φ ηv,w } η We will drop any reference to o if it is trivial. Remark . Note that the above construction depends on a fixed choice of acocycle. Let us fix a vertex v and consider the cocycle ˜ σ given by˜ σ η = σ η if v / ∈ η − σ η if v ∈ η For the construction associated to ˜ σ we keep the fibrations for w = v and weconsider the fibration ˜ π v : M v → ( − S v ) which is the opposite fibration to π v butthe orientation of M v remains unchanged. As a consequence ˜ φ ηv = ( φ ηv ) − and˜ µ ηv = ( µ ηv ) − , when v ∈ η . Note that ˜Φ ηv,w = Φ ηv,w and the resulting manifold is thesame as above. Hence, by Remark 2.1, the manifold depends only on o and noton the particular choice of a representative cocycle. Example 2.3.
Let X be a complex surface and let D = S rj =1 D j be a normalcrossing compact divisor in X . Let Γ be the dual graph of D and define thefunctions g, e as the genus and self-intersection. Then the boundary of a regularneighbourhood of D is homeomorphic to the graph manifold of (Γ , g, e ). If theintersection matrix of D is negative definite then D can be obtained as the excep-tional divisor of a resolution of an isolated surface singularity. That is, the link ofan isolated surface singularity is always a plumbing manifold, whose graph is thedual graph of the resolution. This example is the main motivation for this work.The rest of the paper is devoted to proof that the construction of § Topological constructions
In this section we introduce different constructions which will be used in thesequel.
Drilled bodies.Definition 3.1.
A ( g, n ) -drilled body is a product H g,n := Σ g,n × I , where I := [0 , g,n is an oriented compact surface of genus g and n boundary components,with n > a aa abb bbf ff f (a) ( S ) × I = Σ , × I a aa abb bb (b) H , a aa abbbb (c) H , Figure 3.1. Products
Lemma 3.2.
The boundary of a ( g, n ) -drilled body is an oriented surface of genus g + n − which is decomposed as a union of two copies of Σ g,n and n cylinders,called the drill holes .Proof. It is clear that ∂H g,n is an oriented surface for being the boundary of anoriented 3-manifold. It can be decomposed as follows: ∂H g,n = Σ g,n × { , } ∪ ( ∂ Σ g,n × I ) . Since n >
0, the surface is connected. Its Euler characteristic is: χ ( ∂H g,n ) = 2 χ (Σ g,n ) = 2(2 − g − n ) = 2 − g + n − . (cid:3) Theorem 3.3. A ( g, n ) -drilled body is a (2 g + n − -handle body.Proof. We consider Σ g,n as the closure of the complement of n pairwise disjointdisks in a closed surface Σ g of genus g . This surface is represented as a 4 g -polygon P g with the usual identifications; recall all the vertices are identified as a point P .The first disk to be removed can be chosen with center at P . The other n − P g . EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 11
Hence the surface Σ g,n can be seen as an 8 g -gone Q n − g (with n − D , . . . , D n in its interior), with identifications in 4 g of its edges. Recall that ∂D is obtained by gluing the non-identified edges of Q n − g .We can choose n − α j joining ∂D j and ∂D , j = 2 , . . . , n . Note that if we cut along these segments and the identified edges,we obtain a topological disk.The 3-manifold H g,n can be seen as a drilled prism with basis Q n − g , where thevertical faces are identified as the corresponding edges on Q n − g .Let us cut Q n − g × I along the 2 g identified faces and the n − α j × I . We obtain the product of a disk and an interval which is a topological3-ball. (cid:3) Float gluings.
We are going to define another construction. Let M be an oriented 3-manifoldwith boundary and let η be an oriented simple closed curve in ∂M ; then a regularneighbourhood C in ∂M of η is an annulus. Consider an oriented solid torus V withoriented core γ and let ˜ γ be a longitude in ∂V . Let A be tubular neighbourhoodof ˜ γ in ∂V ; note that A is an annulus. Let ψ : C → A be an orientation-reversinghomeomorphism. Proposition 3.4.
The manifold M ∪ ψ V is homeomorphic to M .Proof. The solid torus V can be retracted to A and this rectraction induces anisotopy between M ∪ ψ V and M . (cid:3) Remark . Note that in the previous construction there are two possible choicesfor the gluing morphism ψ . One of them identifies γ with η and the other one, γ with η − . Moreover, the gluings of the boundary components of C and A areinterchanged. Definition 3.6.
The above operation is called a float gluing of M along C . Definition 3.7.
Given a handle-body M of genus g we say that a simple closedcurve γ ⊂ ∂M is a float curve if there is a cutting system of curves in ∂M suchthat γ intersects this system in only one point, and this intersection is transverse. Example 3.8.
Let us consider a solid torus V and let γ be a simple closed curvein ∂V isotopic to the core of V . Let V g − be a handle-body of genus g −
1. Let V g be the handle-body obtained by gluing two disks in the boundaries of V and V g − ; for further use, we will refer to this operation as the handle sum of V and V g − ; we can assume that γ is disjoint with the disk in ∂V used for the handlesum. Then γ ⊂ V g is a float curve of V g since it cuts only the meridian of V . Definition 3.9.
The pair ( V g , γ ) is called a standard float-curve system of genus g . Lemma 3.10.
Let M be a handle-body of genus g and let γ ⊂ M be a floatcurve. Then, the pair ( M, γ ) is homeomorphic to a standard float-curve system ofgenus g .Proof. A handle-body can be seen as a closed ball B with 2 g pairwise disjoint disksin the boundary glued in pairs. In this model a curve γ is a segment joining a pairof glued disks and avoiding the other ones. Two such models can be connected bya homeomorphism. (cid:3) Proposition 3.11.
Let M , M be two handle-bodies of genus g , g ≥ . Fixfloat curves γ , γ in each one and consider regular neighbourhoods A , A of thesecurves in ∂M , ∂M , respectively. Let ψ : A → A be an orientation-reversinghomeomorphism. Then, M ∪ ψ M is a handle-body of genus g + g − .Remark . If M is of genus 1 the above operation is a particular case of floatgluing since we only need the curve in the solid torus to be a float curve. In fact,the above proposition remains true if we only ask γ to be a float curve, but wedo not use this more general fact. Proof.
Note that M can be constructed as a handle sum of a solid torus V anda handle body of genus g −
1. This operation can be performed in order to have A ⊂ V and γ homotopic to the core of V .Then, the gluing of M and M can be performed as a float gluing followed bya handle sum. (cid:3) Remark . In fact, we can be more specific with the handlebody structure of M := M ∪ ψ M . Consider a system of cutting curves α , . . . , α g for M and β , . . . , β g for M . We first assume that only α (resp. β ) intersects γ (resp. γ ), at only one point and transversally (which is possible since γ and γ arefloat curves). Let ˇ α be the piece of α outside the small neighbourhood of γ used for the gluing; define ˇ β accordingly. We can isotopically move β such that δ := ˇ α · ˇ β is a cycle in the boundary of M . Then, δ, α , . . . , α g , β , . . . , β g is acutting system for M . Remark . This process can be generalized when α i , 1 ≤ i ≤ h , interesects γ transversally at one point and α i ∩ γ = ∅ if h < i ≤ g and a similar fact arisesfor the other system for some h . In this case we can choose suitable curves α ′ i ,2 ≤ i ≤ h (resp. β ′ i , 2 ≤ i ≤ h ) parallel to α (resp. β ) such that ˜ β i := ˇ α ′ i · ˇ β i ,2 ≤ i ≤ h (resp. ˜ α i := ˇ α i · ˇ β ′ i , 2 ≤ i ≤ h ) are cycles. Then(3.1) δ, ˜ α , . . . , ˜ α h , α h +1 , . . . , α g , ˜ β , . . . , ˜ β h , β h +1 , . . . , β g EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 13 is a cutting system of M . We can prove it using handle-slide moves of α i (resp. β i ), 1 < i ≤ h (resp. h ), along α (resp. β ) in order to pass to the situationof Remark 3.13; after the construction of the cutting system of M we perform inverse handle-slide moves along δ and we recover the system (3.1).The same idea can be used if we identify two different annuli in a single handlebody. Proposition 3.15.
Let M be a handle body of genus g ≥ . Fix a cutting systemof curves and two disjoint float curves γ , γ such that they intersect differentcurves of the cutting system α , α . Consider regular neighbourhoods A , A of γ and γ respectively. Let ψ : A → A be an orientation-reversing homeomorphism.Then the quotient M ψ of M by ψ is a handle body of genus g .Proof. Up to homeomorphism, we may assume that the float systems are standardones. In that case, M is a handle sum of solid tori, being γ and γ the longitudesof two of them. The identification then produces a float gluing bewteen these twosolid tori, so we obtain again a handle sum of solid tori, but introducing a loop inthe chain of handle sums. This loop introduces a new handle, that compensatesthe one lost by the identification. (cid:3) α γ γ α β ˜ α ˜ α Figure 3.2. Gluing disjoint float curves in a handle body
Remark . In Figure 3.2 it can be seen how a new cutting curve is obtainedby joining the two identified ones, and another one appears for the handle corre-sponding to the cycle. We are going to check that the latter corresponds to thecommutator of α and γ .Let F = ∂M and consider regular neighbourhoods N ( γ ) and N ( γ ) of γ and γ bounded by four curves γ ± i . The surface F ψ := ∂M ψ is obtained as follows. Consider the quotient of F \ ( N ( γ ) ∪ N ( γ )) obtained by gluing γ +1 with γ − and γ − with γ +2 in order to obtain an oriented 3-manifold.Note that F = ∂M and F ψ = ∂M ψ are equal outside regular neighbourhoodsof γ and γ . A cutting system for M ψ can be constructed as follows. We keepthe curves α , . . . , α g of the cutting system of M and we add two new curves ˜ α and ¯ α . The curve ¯ α is the connected sum α α obtained as the union of twopieces ˇ α , ˇ α as δ in Remark 3.13. The curve ¯ α is the image by the gluing of thecurve β in M ,which is the commutator of α and γ (see Figure 3.2). Note thatthe commutator of α and γ could also be chosen instead of β .4. Heegaard splittings of S -bundles over surfaces withunimodular Euler number Let π : M → S be an oriented S -bundle over a closed oriented surface S ofgenus g , with Euler number e ∈ Z ≡ H ( S ; Z ). Consider a small closed disk D ⊆ S and consider the surface with boundary ˇ S := S \ D . Since H ( ˇ S ; Z ) is trivial, thereexists a section s : ˇ S → M of π . We take another parallel section s . These twosections divide ˇ M = π − ( ˇ S ) in two pieces M and M ; which are oriented compact3-manifolds with boundary, and satisfy that M ∩ M = ∂M ∩ ∂M = S ` N ,where S := s ( ˇ S ) and N := s ( ˇ S ). We will now show how to use these two piecesto construct a Heegaard splitting of M when the Euler number of the fibrationis e = ± v , g v = g , e v = ± Convention 4.1.
Once the two sections s , s have been chosen, we choose M and M in such a way that the orientations on N induced by M and s coincide.This means that a positive half-fiber inside M goes from S to N .The boundary of M is obtained by gluing S and N with an annulus C whichfibers over ∂D = ∂ ˇ S (whose fibers are positive half-fibers inside M homeomorphicto [0 , ∂M = S ∪ C ′ ∪ N , where C ′ is the other annulus in M . Note that C ∪ C ′ is the torus π − ( ∂ ( D )) = ∂π − ( D ) ( C and C ′ have commonboundaries). Proposition 4.2.
The -manifolds M , M are g -handle bodies.Proof. This manifold is, by construction, the drilled body H g, , see Theorem 3.3.Since M is homeomorphic to M , it is also a 2 g -handle body. (cid:3) Theorem 4.3.
Let ˜ M := M ∪ π − ( D ) . The manifold ˜ M is homeomorphic to M and, hence, it is a g -handle body. EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 15
Proof.
Note that C ′ is the annulus along which M and π − ( D ) are glued. Let K be the core of this annulus. Since e = ± K is homologous to the core of π − ( D )and the statement follows from Proposition 3.4. (cid:3) Corolary 4.4.
The submanifolds M and ˜ M form a Heegaard splitting of M ofgenus g . Heegaard diagram of a unimodular S -bundle. Let us denote Σ := ∂M = ∂ ˜ M (oriented as boundary of M ), which is thegluing of S , N and the cylinder C ∼ = ∂D × I . Note that N inherits the orientationof S while S inherits the opposite one. p p q q b ν λ ν ′ λ a = ν · λ · ν ′ · λ γ ′ SN Cγ (a) Cutting curves for M . b ′ a ′ (b) Cutting curves for ˜ M , e = 1. Figure 5.1In this situation, the system of cutting curves for M is formed by two familiesof curves: • Curves a , . . . , a g coming from half of the identified faces in the prysm,see Figure 3.1(b). They are decomposed into four pieces as follows, seeFigure 5.1(a). Consider points p i , p i in C ∩ N and points q i , q i in C ∩ S such that there are half-fibers λ i (from q i to p i ) λ i (from p i to q i ). Pickup a path ν i in S from p i to p i which turns around the i ’th handle like its meridian. We construct a path ν ′ i in N in a similar way with reversedorientation. Then, a i := ν i · λ i · ν ′ i · λ i . It is possible to choose these cyclesto be pairwise disjoint. • Curves b , . . . , b g coming from the other half of the identified faces. Theyare constructed in the same way as the a i , but instead of taking ν i and ν ′ i ,we take paths that turn around the handles like their longitudes. Thesepaths are chosen in such a way that they don’t intersect each other andthey are also disjoint to the paths a i ’s.The prysm of Figure 3.1(b) shows how to prove that this is a system of cuttingcurves.In order to obtain a system of cutting curves for ˜ M we recall its construction.We start with M (homeomorphic copy to M ) which is constructed in the sameway as M but using the other cylinder C ′ . Recall that the union of the twocylinders C and C ′ along their common boundary yields the torus T := π − ( ∂D ),the boundary of the solid torus π − ( D ). So the construction of the system ofcutting curves for M will mimic the one for M replacing the cylinder C by C ′ .Since ˜ M = M ∪ π − ( D ), let us consider the situation at π − ( D ). In order tofix the orientations, we assume that e = 1, leaving the case e = − π − ( D ) is represented as a cylinder whose bottom and top are gluedby a vertical translation in Figure 5.2(a). Note that π − ( D ) is the solid torus usedin the float gluing in order to obtain ˜ M from M .In the torus T , we fix the product structure with oriented section µ (the bound-ary of a disk in the solid torus) and with oriented fibre φ . Let us fix one cuttingcurve ( a i or b i ) of M ; it intersects the cylinder C in two half-fibers. Let λ bethe one from S to N ; let λ ′ be the other half of the fiber in C ′ (which is part ofa cutting curve in M ) but with opposite orientation, in order to go again from S to N ; i.e., λ · λ ′ − is homologous to φ in T .The cylinders C and C ′ have as common boundaries two cycles γ ⊂ N and γ ′ ⊂ S , oriented as boundaries of these surfaces; in Figure 5.2(a), the front partof C is coloured. The homology class of γ in T is (with multiplicative notation) µ − · φ − e (recall e = 1 in Figure 5.2(a)), since the definition of Euler numberimplies that γ · µ · φ e is trivial.The cycle ( γ ′ ) − · ( λ · λ ′ − ) e ∼ γ · φ e ∼ µ − bounds a disk in π − ( D ). The unionof this disk with the cutting disk of M containing λ ′ − in its boundary provides anew disk where λ ′ − is no more in its boundary. If we repeat this process with theother half-fiber in the cutting curve, we obtain the corresponding cutting curvein ˜ M where the half-fibers have been replaced by curves in C ⊂ ∂ ˜ M = ∂M . It EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 17 can be checked that the retraction seen in Proposition 3.4 sends γ ′− e λ to λ ′ andhence this construction provides the cutting curve for ˜ M .Figure 5.1(b) shows the cutting curves of ˜ M for g = 1, e = 1. Note that theblue curves in C turn around as γ when going from S to N . The closed curve γ ′ is oriented as boundary of N and γ is parallel to γ ′ .It is clear that in the case of e = −
1, the same thing will happen but instead ofturning as γ , the curves will turn as γ − , see Figure 5.2(b). λ γ λ ′ φ γ ′ µ (a) From M to ˜ M , e = 1. (b) Example for the case g = 1, e = − Figure 5.26.
Heegaard splittings of arbitrary S -bundles over surfaces In order to construct a Heegaard splitting for arbitrary Euler number e weproceed as follows. Let now ˇ S := S \ S nj =1 D i , where D , . . . , D n are pairwisedisjoint closed disks in S . As before, let s , s : ˇ S → M be arbitrary parallelsections of π . For each j = 1 , . . . , n , let γ j := s ( ∂D j ) (oriented as part of ∂ ˇ S )and let µ j be the boundary of a meridian disk of the solid torus π − ( D j ). By thechoice of orientations the cycle γ j · µ j · φ e j is trivial in H ( π − ( ∂D j ); Z ), for some e j ∈ Z , where φ is an oriented fiber of π . The following is a classical result. Lemma 6.1.
With the above notations, e = P nj =1 e j . Moreover, for every choiceof the e j ’s satisfying this equality, there exists a choice of sections that realizes it. As we did in §
4, we may decompose ˇ M := π − ( ˇ S ) in two pieces M and M ; M and M are oriented compact 3-manifolds with boundary and M ∩ M = ∂M ∩ ∂M = s ( ˇ S ) ` s ( ˇ S ) with the same orientation convention. From Theorem 3.3,the manifolds M and M are (2 g + n − Let us assume that e j = ± j = 1 , . . . , n . Note that M is homeomorphic to M and hence, it is also a (2 g + n − M := M ∪ S nj =1 π − ( D j ).Following the arguments in the proof of Theorem 4.3, we can see that M ∼ = ˜ M and M and ˜ M have the same boundary. We have proven the following result. Theorem 6.2.
The submanifolds M and ˜ M form a Heegaard splitting of M . If e = 0 , a decomposition of this kind of genus g + 1 can be always obtained; and if e = 0 , one of genus g + | e | − .Remark . In this process, we have glued all the solid tori π − ( D j ) to M . Thisis not essential for the proof: we could have glued some of them to M and theresult would be equally valid.Let us describe the cutting curves of M . First, we consider the cutting curvesof §
5. Second, we add curves c j , j = 2 , . . . , n as follows. Consider the paths α j (as in the proof of Theorem 3.3) joining p j ∈ ∂D and q j ∈ ∂D j ; recall that bycutting along them ˇ S becomes a disk. The boundaries c j = s ( α j ) · ( { q j } × I ) · s ( α j ) − · ( { p j } × I ) − . of α j × I , together with the curves of §
5, form a system of cutting curves for M .Following the arguments in §
5, the curves of M mimic the ones of M exceptfor the modification in the cylinders ∂D i × I , 1 ≤ i ≤ n , due to the float gluingof the solid tori π − ( D i ). By the same reasoning as before, these modificationsconsist on a Dehn twist along each cylinder. The orientation of each Dehn twistdepends on the sign of each e i . Note that the cylinder ∂D × I plays a specialrole; it will be called main cylinder . Example 6.4.
Figure 6.1(a) shows this construction for the case of genus zeroand Euler number equal to 3. We choose three solid tori and sections with e i = 1.The resulting Heegaard decomposition has genus 2 and therefore is not minimal,since the manifold in question is a lens space, and as such admits a genus onedecomposition. Figure 6.1(b) shows an example of this construction for the caseof g = 1 , e = 2.7. Heegaard splitting of a plumbed graph manifold with an edge
Let M be a plumbed graph manifold with an edge and two vertices. Thismanifold is obtained as follows. We start with two manifolds W and W , whichare oriented S -bundles π i over closed surfaces S i of genus g i and Euler numbers e i , i = 1 ,
2. We take closed disks D i, ⊂ S i and choose a system of curves µ i , φ i on EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 19 (a) Example for the case g = 0, e = 3. a b c (b) Example for the case g = 1, e = 2. Figure 6.1 π − i ( ∂D i, ) as follows: the curve φ i is an oriented fiber of π i , and µ i is the orientedboundary of a meridian disk of π − i ( ∂D i, ).Then, M is obtained by gluing π − ( S \ D , ) and π − ( S \ D , ) along theirboundaries. These boundaries are tori π − i ( ∂D i, ), i = 1 ,
2, and the gluing is de-scribed by a matrix in GL(2; Z ) once ordered integral bases in H ( π − i ( ∂E i ); Z ) arechosen. For the choice of ( µ , , φ ) and ( µ , , φ ) the matrix is ± ( ), dependingon the sign of the edge as described in §
2. Since the edge is contractible, thecohomology class o of § D j, , . . . , D j,n j ⊂ S j \ D j, , j = 1 , S j := S \ S n j i =0 D j,i . We consider two parallel sections s j, , s j, : ˇ S j → M j of π j as in the previous section.As in §
6, we denote γ j,i := s ( ∂D j,i ) (oriented as part of ∂ ˇ S j ); let µ j,i be theboundary of a meridian disk of π − ( D j,i ). As in that section, we collect the integers e j,i appearing in the equalities (in homology of the boundary tori) γ j,i · µ j,i · φ e j,i j = 1,where φ j is a fiber of π j , and they must satisfy n j X i =0 e j,i = e j . We impose the following conditions: • min { n , n } ≥ • | e j,i | = 1; • ε := e , = e , , determining the sign of the edge. • ∂D i, × I is not a main cylinder.In this case, we can construct Heegaard splittings M i , ¯ M i of W i as in Section 6using the systems of disks { D j, , . . . , D j,n j } . To do the plumbing, we have toremove π − i ( ˚ D i, ) from ¯ M i , but as we saw before, this operation doesn’t changethe topology (since it is the inverse of a float gluing). Let’s denote by ¯ M ′ i theresult of the removal of π − i ( ˚ D i, ) from ¯ M i .Note that after the plumbing, µ , is identified with φ ε , and µ ε , is identifiedwith φ . This implies that γ , and γ , are homologous after the plumbing (becauseof the choice of the edge sign). In particular, it means that we can choose thesections s j,i in such a way that s ,i ( ∂D , ) is identified with s ,i ( ∂D , ). Thisway, the two Heegaard splittings are compatible, and we can extend them to adecomposition of M .Sumarizing, we have now the following decomposition:(7.1) M = (cid:0) M ∪ M (cid:1) [ (cid:16) ¯ M ′ ∪ ¯ M ′ (cid:17) . Proposition 7.1.
The manifolds M ∪ M and ¯ M ′ ∪ ¯ M ′ are handle bodies, i.e.,the decomposition is a Heegaard splitting of M .Proof. It is enough to prove it for M ∪ M . We have already seen that both M and M are handle-bodies. We will show now that they are glued as inProposition 3.11. In order to do so, we have to see that they are glued alongannuli that are neighborhoods of a float curve.Let us consider the torus π − i ( ∂D , ) as the product of µ , and φ . The curves s ,i ( ∂D , ) are parallel curves that meet φ transversally at only one point. Let A = M ∩ π − i ( ∂D , )be the annulus along which the gluing is made. This annulus is a neighborhoodof a curve parallel to s ,i ( ∂D , ).From the construction in Section 6, we see that φ ∩ M is part of a cuttingcurve of M . And moreover, its the only intersection of a cutting curve with thetorus π − i ( ∂D , ).So the annulus A is a regular neighborhood of a float curve in M . Analogously, A is also a float curve in M . By Proposition 3.11, we get the result. (cid:3) It is time now to describe a Heegaard diagram, i.e., to understand what happenswith the cutting curves during the plumbing. Let us consider the cylinders A ⊂ M and A ⊂ M which are identified by the plumbing. EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 21
Let us fix a cutting curve λ of M which intersects once the core of A (a floatcurve). In the neighborhood of A , this curve is decomposed in three connectedcomponents λ b , λ c , λ e where λ c is the part of λ that lies in A . As in §
5, thepath λ c is a half of the fiber φ . Analogously, the cutting curve λ in M ina neighbourhood of A can be divided in three connected components λ b , λ c , λ e .The path λ c is equivalent to a half of the fiber φ and recall that φ is identifiedwith a section µ . γ φ γ µ λ c λ b λ e λ c λ b λ e (a) Gluing of M i , e i, = 1. γ φ γ µ λ c λ b λ e λ c λ b λ e (b) Gluing of M ′ i , e i, = 1. Figure 7.1Let us decompose γ = λ γ · λ ′ γ in two halves where λ γ is the bottom part inFigure 7.1(a). If e i, = 1, we can check that λ c can be isotoped inside A to ( λ c ) − followed by ( λ γ ) − , see Figure 7.1(a). That means that the new cutting curve ¯ λ has two connected components near A ≡ A ; one is λ b · λ e , and the other one is λ b · ( λ γ ) − · λ e .We perform a similar argument for the gluing of M ′ and M ′ . In this casewe consider the other annuli A ⊂ M ′ and A ⊂ M ′ which become identified;they are the other parts of the plumbing tori. Let us choose cutting curves λ ′ , λ ′ which go parallel near the annuli to λ , λ ; in order to emphasize it, we keep theabove notation for their decomposition in the neighborhood of the annuli, seeFigure 7.1(b). Assuming again e i, = 1, we see that λ c can be isotoped inside A to λ c followed by λ ′ γ ; note that the isotopy is done in the back part of A if Figure 7.1(b). The new cutting curve ¯ λ has two connected components near A ≡ A ; one is λ b · λ e , as before, and the other one is λ b · λ ′ γ · λ e . As we see in Figures 7.1(a) and 7.1(b), some of the ends do not fit; in order forthem to fit we have to do a half-turn around γ in the suitable direction. Sincewe have freedom to choose the product structure in the annulus, this is equivalentto keep the intersection of the red curves as fibers, while the intersection of theblue curves perform a full loop. To be precise, since the curve ¯ λ · (¯ λ ) − equals γ near the plumbing (in homology), for e i, = 1 the curve λ turns as γ (whengoing from the first vertex to the second one), see Figure 7.2. It is easily seen thatit turns as γ − for e i, = − λ ¯ λ γ Figure 7.2. Cutting curves for e i, = 1. Example 7.2.
Figure 7.3 illustrates the case of two vertices with genus zero andboth with Euler number −
2. Note that we take n = n = 1 and e i,j = − g = 0 , e = − Heegaard splittings of arbitrary plumbed graphs
In this section, we consider an arbitrary plumbing graph (Γ , g, e, o ); for theplumbing construction we fix an explicit cocycle representing o , consisting onassigning a sign e η to each edge η .Fix a vertex v with valency d v ; this vertex is associated with a fibration π v : M v → S v ; we choose d v + n v pairwise disjoint closed disks in S v , determiningsolid tori in M v . The first d v disks are assigned to a fixed edge η having v asan endpoint. As in §
7, the first d v disks will have associated numbers e η , and EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 23 the remaining disks numbers e v,j , j = 1 , . . . , n v , such that their absolute valueequals 1, and X v ∈ ∂η e η + n v X j =1 e v,j = e v . In general one of the extra disks will correspond to the main cylinder, hence n v ≥ g v = 0, d v = 2, since in this case themain cylinder plays no special role.If Γ is a tree it is enough to iterate the construction of §
7. Note also that thereis no restriction for the choice of the cocyle.Let us consider now the general case where the graph may have cycles. We startby the choice of a cocycle and a spanning tree, for which we proceed as above. Letus now explain the effect of plumbing along the remaining edges.As we saw in Proposition 3.15, the process is different when the plumbing closesa cycle in the graph, since in that case the gluing process is done between twofloat curves of the same handlebody; specially, the way of constructing cuttingcurve systems changes. Proposition 3.15 proves that this process produces also aHeegaard splitting (where the genus remains unchanged). · · ·· · · · · ·· · ·· · ·· · · · · ·· · ·
Figure 8.1. Float gluing that closes a cycleHow to obtain the cutting curves is explained in Remark 3.16. Figure 8.1describes this process in our case, showing how to obtain the new pair of cutting curves from the ones that existed before the plumbing. The first pair of cutting(red and blue) curves is obtained as in the tree case: they are obtained as connectedsum of the preexistent ones. The second pair of cutting curves is constructed asexplained in Remark 3.16, as the union of two parallel copies of a preexistent curveand the boundaries of the identified annuli.9.
Explicit examples
Let us consider some examples of graph manifolds for which we will give a Hee-gaard splitting. These examples come from links of normal surface singularities.
Example 9.1.
Let M be the link of the A n singularity, which is a lens space L ( n, n − n − , − , − , − , − A n graph . . .. . .. . . Figure 9.2. Heegard diagram of the A n graph.With our method we obtain a genus 1 Heegaard splitting where the two curvesintersect n times.From now we will drop the genus weight if it vanishes. EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 25
Example 9.2.
Let us consider the plumbing manifold associated with a graphwith one vertex and Euler number − n , link of a quotient singularity, i.e. the lensspace L ( n, n − n −
1) +1-blow-ups and one − Example 9.3.
The plumbing manifold of Figure 9.3 is also a lens space L (5 , − − Example 9.4.
The link of the singularity defined by z + x + y = 0 ( E -singularity) is the Poincar´e sphere. Our method provides a Heegaard splitting ofgenus 3, where the central vertex needs four drills (three negative ones). − − − − − − − − E -singularity.It is possible to make a simpler Heegaard splitting. Using +1-blow-ups of [8](and one − − Example 9.5.
The graph manifold of Figure 9.6 is also the link of a normalsurface singularity (which cannot be quasihomogeneous) and admits a Heegaardsplitting of genus 5. − − − − − − References [1] J. Bodn´ar and A. N´emethi,
Lattice cohomology and rational cuspidal curves , Math. Res.Lett. (2016), no. 2, 339–375. MR 3512889[2] M. Boileau and J.-P. Otal, Scindements de Heegaard et groupe des hom´eotopies des petitesvari´et´es de Seifert , Invent. Math. (1991), no. 1, 85–107.
EEGAARD SPLITTINGS OF GRAPH MANIFOLDS. 27 [3] M. Boileau and H. Zieschang,
Heegaard genus of closed orientable Seifert -manifolds , In-vent. Math. (1984), no. 3, 455–468.[4] P. Heegaard, Sur l’“Analysis situs” , Bull. Soc. Math. France (1916), 161–242.[5] T. L´aszl´o and A N´emethi, Reduction theorem for lattice cohomology , Int. Math. Res. Not.IMRN (2015), no. 11, 2938–2985. MR 3373041[6] Y. Moriah and J. Schultens,
Irreducible Heegaard splittings of Seifert fibered spaces are eithervertical or horizontal , Topology (1998), no. 5, 1089–1112.[7] A. N´emethi, Links of rational singularities, L-spaces and LO fundamental groups , Invent.Math. (2017), no. 1, 69–83. MR 3698339[8] W.D. Neumann,
A calculus for plumbing applied to the topology of complex surface sin-gularities and degenerating complex curves , Trans. Amer. Math. Soc. (1981), no. 2,299–344.[9] P. Ozsv´ath and Z. Szab´o,
Holomorphic disks and three-manifold invariants: properties andapplications , Ann. of Math. (2) (2004), no. 3, 1159–1245.[10] ,
Holomorphic disks and topological invariants for closed three-manifolds , Ann. ofMath. (2) (2004), no. 3, 1027–1158.[11] D. Rolfsen,
Knots and links , Mathematics Lecture Series, no. 7, Publish or Perish, Inc.,Berkeley CA, 1970.[12] S. Sarkar and J. Wang,
An algorithm for computing some Heegaard Floer homologies , Ann.of Math. (2) (2010), no. 2, 1213–1236.[13] J. Schultens,
Heegaard splittings of graph manifolds , Geom. Topol. (2004), 831–876.[14] F. Waldhausen, Eine klasse von -dimensionalen mannigfaltigkeiten I , Invent. Math. (1967), 308–333.[15] , Eine klasse von -dimensionalen mannigfaltigkeiten II , Invent. Math. (1967),87–117. Departamento de Matem´aticas-IUMA, Universidad de Zaragoza, Campus PlazaSan Francisco s/n, E-50009 Zaragoza SPAIN
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