General tête-à-tête graphs and Seifert manifolds
GGENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS
PABLO PORTILLA CUADRADO
Abstract.
Tête-à-tête graphs and relative tête-à-tête graphs were introduced by N. A’Campo in2010 to model monodromies of isolated plane curves. By recent work of Fdez de Bobadilla, PePereira and the author, they provide a way of modeling the periodic mapping classes that leavesome boundary component invariant. In this work we introduce the notion of general tête-à-têtegraph and prove that they model all periodic mapping classes. We also describe algorithms thattake a Seifert manifold and a horizontal surface and return a tête-à-tête graph and vice versa.
Contents
1. Introduction 1Acknowledgments 22. General tête-à-tête structures 23. Seifert manifolds and plumbing graphs 83.1. Seifert manifolds 83.2. Plumbing graphs 94. Horizontal surfaces in Seifert manifolds 11Handy model of a Seifert fibering. 135. Translation Algorithms 155.1. From general tête-à-tête graph to star-shaped plumbing graph 155.2. From star-shaped plumbing graph to tête-à-tête graphs. 166. Examples 17References 211.
Introduction
In [A’C10] N. A’Campo introduced the notion of pure tête-à-tête graph in order to model mon-odromies of plane curves. These are metric ribbon graphs Γ without univalent vertices that satisfya special property called the tête-à-tête property. One usually sees the ribbon graph as a strongdeformation retract of a surface Σ with non-empty boundary, which is called the thickening . Thetête-à-tête propety says that if you pick a point p , then walk distance of π in any direction from thatpoint and you always turn right at vertices, you get to the same point no matter the inital direction.This property defines an element in the mapping class group MCG + (Σ , ∂ Σ) which is freely periodic.In [Gra15], C. Graf proved that if one allows univalent vertices in tête-à-tête graphs, then theset of mapping classes produced by tête-à-tête graphs are all freely periodic mapping classes of MCG + (Σ , ∂ Σ) with positive fractional Dehn twist coefficients. In [FdBPPPC17] this result wasimproved by showing that one does not need to enlarge the original class of metric ribbon graphsused to prove the same theorem.A bigger class of graphs was introduced in [A’C10], the relative tête-à-tête graphs. These are pairs (Γ , A ) formed by a metric ribbon graph Γ and a subset A ⊂ Γ which is a collection of circles. Seenas a strong deformation retract of a surface, this pair is properly embedded, i.e. (Γ , A ) (cid:44) → (Σ , ∂ Σ) and ∂ Σ \ A (cid:54) = ∅ . They satisfy the relative tête-à-tête property which is similar to the tête-à-têteproperty and defines an element in MCG + (Σ , ∂ Σ \ A ) which is freely periodic. In [FdBPPPC17]it was proved that the set of mapping classes modeled by relative tête-à-tête graphs are all freelyperiodic mapping classes of MCG + (Σ , ∂ Σ \ A ) with positive fractional Dehn twist coefficients at theboundary components in ∂ Σ \ A . Author supported by SVP-2013-067644 Severo Ochoa FPI grant and by project by MTM2013-45710-C2-2-P, thetwo of them by MINECO; also supported by the project ERCEA 615655 NMST Consolidator Grant. a r X i v : . [ m a t h . G T ] D ec PABLO PORTILLA CUADRADO
At this point there is a natural question which was already posed in [Gra14], how can one comple-ment the definition of tête-à-tête graph to be able to model all periodic mapping classes? (even if theydo not leave any boundary component invariant). To cover these cases, we introduce general tête-à-tête graphs (see Definition 2.4). These are metric ribbon graphs with some special univalent vertices
P ⊂ ∂ Σ and a permutation acting on these vertices. An analogous general tête-à-tête property isdefined. A general tête-à-tête graph defines a periodic mapping class in MCG + (Σ) .As our main result, we prove: Theorem A.
The mapping class of any periodic automorphism φ : Σ → Σ of a surface can berealized via general tête-à-tête graphs. Moreover, the general tête-à-tête graph can be extended to apure or relative tête-à-tête graph, thus realizing the automorphism as a restriction to Σ of a periodicautomorphism on a surface ˆΣ ⊃ Σ that leaves each boundary component invariant. The mapping torus of a periodic surface automorphism is a Seifert manifold and a orientablehorizontal surface of a fiber-oriented Seifert manifold has a periodic monodromy induced on it. Hence,it is natural to assign a tête-à-tête graph to a Seifert manifold and a horizontal surface on it and viceversa. The rest of the work is devoted to understanding this relation.In Section 3 we briefly review the theory of Seifert manifolds and plumbing graphs. The theoryof Seifert manifolds is classical and there is plenty of literature about it (see for example [Neu81],[NR78], [Neu97], [JN83], [HNK71], [Hat07] or [Ped09]). Because of this, we try to avoid repeatingwell-known results. However, there is not such thing as standard conventions in Seifert manifolds.Since the conventions that we choose are very important for Section 5, we take some time to fix themcarefully.In Section 4, we review the theory about horizontal surfaces in a Seifert manifold M . This hasbeen studied from different point of views in the literature. For example, in [EN85] it is proved aclassification in the more general case when M is an integral homology sphere. In [Pic01] Pichonprovides existence of fibrations of any graph manifold M by producing algorithmically a completelist of the conjugacy and isotopy invariants of the automorphisms whose associated mapping torus isdiffeomorphic to M . We review some of this results and write them in a language that best suits ournotation and conventions. Among these results is a classification of horizontal surfaces of a Seifertmanifold with boundary.In Section 5 we detail two algorithms. One takes a Seifert manifold and a horizontal surface asinput and returns as output a general, relative or pure tête-à-tête graph realizing the horizontalsurface and its monodromy. This algorithms differs from similar results in the literature in that ourmethod produces directly the monodromy (in this case the tête-à-tête graph) without computing theconjugacy and isotoy invariants of the corresponding periodic mapping class. The other algorithmworks in the opposite direction by taking a general, relative or pure tête-à-tête graph and producingthe corresponding Seifert manifold and horizontal surface.Finnaly, Section 6 contains a couple of detailed examples in which we apply the two algorithms. Acknowledgments
I thank my advisors Javier Fdez. de Bobadilla and María Pe Pereira for many useful commentsand suggestions during the writing of this work. I am specially grateful to Enrique Artal Bartolo fortaking the time to explain carefully many of the aspects of the theory of Seifert manifolds.I also thank BCAM for having the ideal environment in which most of this work was developed.2.
General tête-à-tête structures
In this section we study any orientation preserving periodic homeomorphism. Let φ : Σ → Σ be such a homeomorphism. We realize its boundary-free isotopy type and its conjugacy class in M CG (Σ) by a generalization of tête-à-tête graphs, using a technique that reduces to the case ofhomeomorphisms of a larger surface that leave all boundary components invariant.Contrarily to what was done in [FdBPPPC17], we allow ribbon graphs with some special univalentvertice.
Definition 2.1.
A ribbon graph with boundary is a pair (Γ , P ) where Γ is a ribbon graph, and P is theset of univalent vertices, with the following additional property: given any vertex v of valency greaterthan in the cyclic ordering of adjacent edges e ( v ) there are no two consecutive edges connecting v with vertices in P . ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 3
In order to define the thickening of a ribbon graph with boundary we need the following construc-tion:Let Γ (cid:48) be a ribbon graph (without univalent vertices)and let Σ be its thickening. Let g Γ (cid:48) : Σ Γ (cid:48) → Σ be the gluing map. The surface Σ Γ (cid:48) splits as a disjoint union of cylinders (cid:96) i (cid:101) Σ i . Let w be a vertexof Γ (cid:48) . The cylinders (cid:101) Σ i such that w belongs to g Γ (cid:48) ( (cid:101) Σ i ) are in a natural bijection with the pairs ofconsecutive edges ( e (cid:48) , e (cid:48)(cid:48) ) in the cyclic order of the set e ( w ) of adjacent edges to w .Let (Γ , P ) be a ribbon graph with boundary. The graph Γ (cid:48) obtained by erasing from Γ the set E of all vertices in P and its adjacent edges is a ribbon graph. Consider the thickening surface Σ of Γ (cid:48) .Let e be an edge connecting a vertex v ∈ P with another vertex w , let e (cid:48) and e (cid:48)(cid:48) be the inmediatepredecesor and succesor of e in the cyclic order of e ( w ) . By the defining property of ribbon graphswith boundary they are consecutive edges in e ( w ) \ E , and hence determine a unique associatedcylinder which will be denoted by (cid:101) Σ i ( v ) .Each cylinder (cid:101) Σ i has two boundary components, one, denoted by (cid:101) Γ i corresponds to the boundarycomponent obtained by cutting the graph, and the other, called C i , corresponds to a boundarycomponent of Σ . Fix a cylinder (cid:101) Σ i . Let { v , ..., v k } be the vertices of P whose associated cylinderis (cid:101) Σ i . Let { e , ..., e k } be the corresponding edges, let { w , ..., w k } be the corresponding vertices at Γ (cid:48) , and let { w (cid:48) , ..., w (cid:48) k } be the set of preimages by g Γ (cid:48) contained in (cid:101) Σ i . The defining property ofribbon graphs with boundary imply that w (cid:48) i and w (cid:48) j are pairwise different if i (cid:54) = j . Furthermore, since { w (cid:48) , ..., w (cid:48) k } is included in the circle (cid:101) Γ i , which has an orientation inherited from Σ , the set { w (cid:48) , ..., w (cid:48) k } ,and hence also { e , ..., e k } and { v , ..., v k } has a cyclic order. We assume that our indexing respectsit. Fix a product structure S × I for each cylinder (cid:101) Σ i , where S × { } corresponds to the boundarycomponent (cid:101) Γ i , and S × { } corresponds to the boundary component of C i .Using this product structure we can embedd Γ in Σ : for each vertex v ∈ P consider the correspond-ing cylinder (cid:101) Σ i ( v ) , let w (cid:48) be the point in (cid:101) Γ i ( v ) determined above. We embedd the segment g Γ (cid:48) ( w (cid:48) × I ) in Σ .Doing this for any vertex v we obtain an embedding of Γ in Σ such that all the vertices P belongto the boundary ∂ Σ , and such that Σ admits Γ as a regular deformation retract. Definition 2.2.
Let (Γ , P ) be a ribbon graph with boundary. We define the thickening surface Σ of (Γ , P ) to be the thickening surface of Γ (cid:48) toghether with the embeding (Γ , P ) ⊂ (Σ , ∂ Σ) constructedabove. We say that (Γ , P ) is a general spine of (Σ , ∂ Σ) . Definition 2.3 (General safe walk) . Let (Γ , P ) be a metric ribbon graph with boundary. Let σ beapermutation of P .We define a general safe walk in (Γ , P , σ ) starting at a point p ∈ Γ \ v (Γ) to be a map γ p : [0 , π ] → Γ such that γ p (0) = p and | γ (cid:48) p | = 1 at all times. when γ p gets to a vertex of valency ≥ it continues along the next edge in the cyclic order. when γ gets to a vertex in P , it continues along the edge indicated by the permutation σ . Definition 2.4 (General tête-à-tête graph) . Let (Γ , P , σ ) be as in the previous definition. Let γ p , ω p be the two safe walks starting at a point p in Γ \ v (Γ) .We say Γ has the general tête-à-tête property if • for any p ∈ Γ \ v (Γ) we have γ p ( π ) = ω p ( π ) Moreover we say that (Γ , P , σ ) gives a general tête-à-tête structure for (Σ , ∂ Σ) if (Σ , ∂ Σ) is thethickening of (Γ , P ) . In the following construction we associate to a general tête-à-tête graph (Γ , P , σ ) a homeomorphismof (Γ , P ) which restricts to the permutation σ in P ; we call it the general tête-à-tête homeomorphismof (Γ , P , σ ) . We construct also a homeomorphism of the thickening surface which leaves Γ invari-ant and restricts on Γ to the general tête-à-tête homeomorphism of (Γ , P , σ ) . We construct thehomeomorphism on the graph and on its thickening simultaneously.Consider the homeomorphism of Γ (cid:48) \ v (Γ) defined by p (cid:55)→ γ p ( π ) . PABLO PORTILLA CUADRADO
The same proof of [FdBPPPC17, Lemma 3.6] shows that there is an extension of this homeomorphismto a homeomorphism σ Γ : Γ → Γ . The restriction of the general tête-à-tête homeomorphism that we are constructing to Γ (cid:48) coincideswith σ Γ . The mapping σ Γ leaves Γ (cid:48) invariant for being a homeomorphism. Let (cid:101) Γ (cid:48) be the union of thecircles (cid:101) Γ i . The homeomorphism σ Γ | Γ (cid:48) lifts to a periodic homeomorphism ˜ σ : g − (cid:48) ( (cid:101) Γ (cid:48) ) → g − (cid:48) ( (cid:101) Γ (cid:48) ) , which may exchange circles in the following way. For any p ∈ (cid:101) Γ (cid:48) , the points in g − (cid:48) ( p ) correspondsto the starting poing of safe walks in (cid:101) Γ (cid:48) starting at p . A safe walk starting at p is determined by thepoint p and an starting direction at an edge containg p .As we have seen, if (Γ , P , σ ) is a general tête-à-tête structure for (Σ , ∂ Σ) then the surface Σ Γ (cid:48) is a disjoint union of cylinders. The lifting ˜ σ extends to Σ Γ (cid:48) similarly as with the definition of thehomoemorphism corresponding to a tête-à-tête structure defined in [FdBPPPC17]. This extensioninterchanges some cylinders (cid:101) Σ i and goes down to an homeomorphism of Σ . We denote it by φ (Γ , P ,σ ) .If necessary, we change the embedding of the part of Γ not contained in Γ (cid:48) in Σ such that it is invariantby φ (Γ , P ,σ ) . This is done by an adequate choice of the trivilizations of the cylinders. Definition 2.5.
The homeomorphism φ (Γ , P ,σ ) is by definition the homeomorphism of the thickening,and its restriction to Γ the general tête-à-tête homeomorphism of (Γ , P , σ ) . With the notation and definitions introduced we are ready to state and proof the main result ofthe work.
Theorem 2.6.
Given a periodic homeomorphism φ of a surface with boundary (Σ , ∂ Σ) which is nota disk or a cylinder, the following assertions hold:(i) There is a general tête-à-tête graph (Γ , P , σ ) such that the thickening of (Γ , P ) is (Σ , ∂ Σ) , thehomeomorphism φ leaves Γ invariant and we have the equality φ | Γ = φ (Γ , P ,σ ) | Γ .(ii) We have the equality of boundary-free isotopy classes [ φ | Γ ] = [ φ (Γ , P ,σ ) ] .(iii) The homeomorphisms φ and φ (Γ , P ,σ ) are conjugate.Proof. In the first part of the proof we extend the homeomorphism φ to a homeomorphism ˆ φ of abigger surface ˆΣ that leaves all the boundary components invariant. Then, we find a tête-à-tête graph ˆΓ for ˆ φ such that ˆΓ ∩ Σ , with a small modification in the metric and a suitable permutation, is ageneral tête-à-tête graph for φ .Let n be the order of the homeomorphism. Consider the permutation induced by φ in the set ofboundary components. Let { C , ..., C m } be an orbit of cardinality strictly bigger than , numberedsuch that φ ( C i ) = C i +1 and φ ( C m ) = C . Take an arc α ⊂ C small enough so that it is disjointfrom all its iterations by φ . Define the arcs α i := φ i ( α ) for i ∈ { , ..., n − } , which are contained in ∪ i C i . Obviously we have the equalities α i +1 = φ ( α i ) and φ ( α n − ) = α = α . α α α α α α a a a a a a A A A A A A Figure 2.1.
Example of a star-shaped piece S with 6 arms on the left and boundary componentscomponents on the right. The arcs along which the two pieces are glued, are marked in red. In blueand red are the boundaries of the two disks that we used to cap off the new boundaries. We consider a star-shaped piece S of n arms as in Figure 2.1. We denote by D the central boundarycomponent. Let a , . . . , a n − be the boundary of the arms of the star-shaped piece labelled in the ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 5 picture, oriented counterclockwise. We consider the rotation r of order n acting on this piece suchthat r ( a i ) = a i +1 . Note that this rotation leaves D invariant.We consider the surface ˆΣ obtained by gluing Σ and S identifying a i with α i reversing the ori-entation, and such that φ and the rotation r glue to a periodic homeomorphism ˆ φ in the resultingsurface.The boundary components of the new surface are precisely the boundary components of Σ differentfrom { C , ..., C m } , the new boundary component D , and the boundary components C (cid:48) ,..., C (cid:48) k thatcontain the part of the C i ’s not included in the union ∪ n − i =0 α i .The homeomorphism ˆ φ leaves D invariant and may interchange the new boundary components C (cid:48) ,..., C (cid:48) k . We cup each component C (cid:48) i with a disk D i and extend the homeomorphism by the Alexandertrick, obtaining a homeomorphism ˆ φ of a bigger surface ˆΣ . The only new ramification points thatthe action of ˆ φ may induce are the centers t i of these disks. We claim that, in fact, each of the t i ’s isa ramification point.Denote the quotient map by p : ˆΣ → ˆΣ ˆ φ . In order to prove the claim notice that the difference ˆΣ \ Σ is homeomorphic to a closed surface with m +1 disks removed. On the other hand the difference of quotient surfaces ˆΣ ˆ φ \ Σ φ is homeomorphic toa cylinder. Since m is strictly bigger than , Hurwitz formula for p forces the existence of ramificationpoints. Since p is a Galois cover each t i is a ramification point.The new boundary component of ˆΣ ˆ φ corresponds to p ( D ) , where D is invariant by ˆ φ . The point q := p ( t i ) is then a branch point of p .We do this operation for every orbit of boundary components in Σ of cardinality greater than . Then we get a surface ˆΣ and an extension ˆ φ of φ that leaves all the boundary componentsinvariant. The quotient surface ˆΣ ˆ φ is obtained from Σ φ attaching some cylinders C j to some boundarycomponents. Let p : ˆΣ → ˆΣ ˆ φ denote the quotient map. Comparing p | Σ and p | ˆΣ , we see that we have only one new branching point q j in every cylinder C j .Now we construct a tête-à-tête graph for ˆ φ modifying slightly the construction of [FdBPPPC17,Theorem 5.12].To fix ideas we consider the case in which the genus of the quotient ˆΣ ˆ φ is positive. The modifi-cation of the genus case is exactly the same. As in [FdBPPPC17, Theorem 5.12] we use a planarrepresentation of Σ φ as a convex g -gon in R with r disjoint open disks removed from its convexhull and whose edges are labelled clockwise like a b a − b − a b a − b − . . . a g b g a − g b − g , we numberthe boundary components C i ⊂ ∂ Σ , ≤ i ≤ r , we denote by d the arc a b a − b − . . . a g b g a − g b − g ,and we consider l ,..., l r − arcs as in Figure 2.2. We denote by c ,..., c r the edges in which a − (and a ) is subdivided according to the component p ( C i ) they enclose.We impose the further condition that each of the regions in which the polygon is subdivided bythe l i ’s encloses not only a component p ( C i ) , but also the branching point q i that appears in thecylinder C i . We assume that the union of d , a b a − b − and the l i ’s contains all the branching pointsof p except the q i ’s.In order to be able to lift the retraction we need that the spine that we draw in the quotientcontains all branching points. In order to achieve this we add an edge s i joining q i and some interiorpoint q (cid:48) i of l i for i = 1 , . . . , r − and joining q r with some interior point q (cid:48) r of l r − . We may assumethat q (cid:48) i is not a branching point. We consider the circle p ( C i ) and ask s i to meet it transversely to itat only point. See Figure 2.3. We consider the graph Γ (cid:48) as the union of the previous segments andthe s i ’s. Clearly the quotient surface retracts to it. Since it contains all branching points its preimage ˆΓ is a spine for ˆΣ . It has no univalent vertices since the q i ’s are branching points of a Galois cover.In order to give a metric in the graph we proceed as follows. We give the segments d and C i ’s thesame length they had in the proof of [FdBPPPC17, Theorem 5.12]. We impose every s i to have lengthsome small enough (cid:15) and the part of s i inside the cylinder C i to have length (cid:15)/ (see Figure 2.3). Wegive each segment l i length L − (cid:15) . It is easy to check that the preimage graph ˆΓ with the pullbackmetric is tête-à-tête . PABLO PORTILLA CUADRADO d c c c r a b a − b − . . . s s s r l r − l l a a a ′ a r l l l r − s s s r Figure 2.2.
Drawing of Γ (cid:48) for the case genus (ˆΣ ˆ φ ) ≥ in the firts image and genus (ˆΣ ˆ φ ) = 0 inthe second. p ( C i ) p ( D ) q := p ( t i ) l s ⊂ Γ ˆ φ ǫ/ ǫ/ Figure 2.3.
Neighbourhood of q = p ( t i ) in ˆΣ ˆ φ and edge s joining q and l . Now we consider the graph
Γ := ˆΓ ∩ Σ with the restriction metric, except on the edges meeting ∂ Σ whose length is redefined to be (cid:15) . Along the lines of the proof of [FdBPPPC17, Theorem 5.12]we get that φ (Γ , P ,σ ) and φ are isotopic and conjugate. If we denote by P the set of univalent verticesof Γ , it is an immediate consequence of the construction that (Γ , P , σ ) with the obvious permutation σ of P is a general tête-à-tête graph with φ (Γ , P ,σ ) | Γ = φ | Γ . (cid:3) Example 2.7.
We show an example that illustrates these ideas. Let Σ be surface of genus and boundary components C , C , C embedded in R as in the picture 2.4. Let φ : Σ → Σ be therestriction of the space rotation of order that exchanges the boundary components. We observethat in particular φ | C i = id for i = 0 , , .We consider the star-shaped piece S with arms together with the order rotation r that exchangesthe arms (see the picture Figure 2.4).We glue S to Σ as the theorem indicates: we mark a small arc α ⊂ C and all its iteratedimages by the rotation. Then we glue α , α , α to a , a , a respectively by orientation reversinghomeomorphisms. We get a new surface ˆΣ := Σ ∪ S with boundary components. We cap theboundary component that intersects C ∪ C ∪ C with a disk D and extend the homeomorphism tothe interior of the disk getting a new surface ˆΣ and a homeomorphism ˆ φ .Using Hurwitz formula − g − − we get that the surface we are gluing to Σ has genus and hence it is a sphere with boundary components. See picture Figure 2.5. Three of them areidentified with C , C , C , and the 4-th is called C and is the only boundary component of ˆΣ . ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 7 π/ Figure 2.4.
On the left, the torus Σ with disks removed and the orbit of an arc marked, thatis, arcs in red. In the center, the star-shaped piece S with arms to be glued to the torus alongthose arcs. On the right, the surface we get after gluing, with boundary components, one of theminvariant by the induced homeomorphism. S S D B ˜ B Figure 2.5.
On the left, the torus with disks removed and the orbit of an arc marked. On theright, the star-shaped piece with arms to be glued to the torus along those arcs. We compute the orbit space ˆΣ ˆ φ by the extended homeomorphism ˆ φ and get a torus with boundarycomponent. We consider the graph Γ (cid:48) as in picture Figure 2.6. We put a metric in this graph. Weset every edge of the hexagon to be π/ − (cid:15)/ long and the path joining the hexagon with the branchpoint to be (cid:15) long. In this way, if we look at the result of cutting ˆΣ ˆ φ along the graph ˆ φ we see that theonly boundary component that maps to the graph by the gluing map has length π/ − (cid:15)/ (cid:15) = π . B φ ( S S D ) ˜ φ ˜ B ˜ φ a a a a − a − a − Figure 2.6.
On the lower part we have the original surface. On the upper part we have the surfacethat we attach, in this case a sphere with holes removed. The preimage ˆΓ of Γ (cid:48) by the quotient map is a tête-à-tête graph whose thickening is ˆˆΣ . Itsassociated homeomorphism ˆ φ leaves Σ invariant and its restriction to it coincides with the rotation φ . Moreover (ˆΓ ∩ Σ , ˆΓ ∩ ∂ Σ) is a general spine of (Σ , ∂ Σ) . Modifying the induced metric in ˆΓ ∩ Σ asin the proof of the Theorem and adding the order cyclic permutation to the valency vertices weobtain a tête-à-tête graph whose associated homeomorphism equals φ . PABLO PORTILLA CUADRADO
12 3 (cid:15)π/ − (cid:15)/ Figure 2.7.
On the left, the torus with disks removed and the orbit of an arc marked. On theright, the star-shaped piece with arms to be glued to the torus along those arcs. Seifert manifolds and plumbing graphs
In this subsection we recall some theory about Seifert manifolds and plumbing graphs and fix theconventions used in this work. For more on this topic, see [Neu81], [NR78], [Neu97], [JN83], [HNK71],[Hat07] or [Ped09]. In many aspects we follow [Ped09].3.1.
Seifert manifolds.
Let p, q ∈ Z with q > and gcd( p, q ) = 1 . Let D × [0 , be a solid cylinder.We consider the natural orientation on D × [0 , .Let ( t, θ ) be polar coordinates for D . Let r p/q : D → D be the rotation ( t, θ ) (cid:55)→ ( t, θ + 2 πp/q ) .Let T p,q be the mapping torus of D induced by the rotation r p/q , that is, the quotient space D × [0 , t, θ, ∼ ( t, r p/q ( θ ) , . If p, p (cid:48) ∈ Z with p ≡ p (cid:48) mod q , then the rotations r p/q and r p (cid:48) /q are exactly the same map so T p (cid:48) ,q = T p,q . The resulting space is diffeomorphic to a solid torus naturally foliated by circles whichwe call fibers . We call this space a solid ( p, q ) -torus or a solid torus of type ( p, q ) . It has an orientationinduced from the orientation of D × [0 , ⊂ R . The torus ∂T p,q is oriented as boundary of T p,q .We call the image of { (0 , } × [0 , ⊂ D × [0 , in T p,q the central fiber . We say that q is the multiplicity of the central fiber. If q = 1 we call the central fiber a typical fiber and if q > we callthe central fiber a special fiber . Also any other fiber than the central fiber is called a typical fiber.If a and b are two closed curves in ∂T p,q , let [ a ] · [ b ] denote the oriented intersection number oftheir classes in H ( ∂T p,q ; Z ) . We describe classes of simple closed curves on H ( ∂T p,q , Z ) :(1) A meridian curve m := ∂D × { y } . We orient it as boundary of D × { y } .(2) A fiber f on the boundary ∂T p,q . We orient it so that the radial projection on the centralfiber is orientation preserving. It satisfies that [ m ] · [ f ] = q .(3) A longitude l is a curve such that [ l ] is a generator of H ( T p,q ; Z ) and [ m ] · [ l ] = 1 .(4) A section s . That is a closed curve that intersects each fiber exactly once. It is well definedup integral multiples of f . It is oriented so that [ s ] · [ f ] = − . f lm Figure 3.1.
A torus T , with some closed curves marked on its boundary. In orange a fiber f , inblue a meridian m and in red a longitude l . ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 9
We have defined two basis of the homology of ∂T p,q , so we have that there must exist unique a, b ∈ Z such that the equation(3.1) ([ s ][ f ]) = ([ m ][ l ]) (cid:18) a pb q (cid:19) holds in H ( ∂T p,q ; Z ) . The matrix is nothing but a change of basis.The matrix of the equation has determinant − because [ s ] , [ f ] is a negative basis in the homologygroup H ( ∂T p,q ; Z ) . Therefore bp ≡ q . Changing the class [ s ] by adding integer multiples of [ f ] to it, changes b by integer multiples of q .We now fix conventions on Seifert manifolds. Let B (cid:48) be an oriented surface of genus g and r + k boundary components, M (cid:48) := B (cid:48) × S and s (cid:48) : M (cid:48) × S → B (cid:48) the projection onto B (cid:48) . Let ( α , β ) , . . . , ( α k , β k ) be k pairs of integers with α i > for all i = 1 , . . . , k . Let N , . . . N k be k boundary tori on M (cid:48) . On each of them consider the following two curves s i := B (cid:48) × { } ∩ N i and anyfiber f i ⊂ N i . Orient them so that { [ s i ] , [ f i ] } is a positive basis of N i as boundary of M (cid:48) . For each i , consider a solid torus T i = D × S and the curves m = ∂D × { } and l := { pt } × S oriented sothat { [ m ] , [ l ] } is a positive basis of T i . Attach T i to N i along its boundary by (cid:0) − α i c − β i d (cid:1) : ∂T i → N i with respect to the two given basis. The numbers c and d are integers such that the matrix hasdeterminant − . Note that, since the first column defines the attaching of the meridian, the gluingis well defined up to isotopy.The foliation on N i extends to all T i and gives it a structure of a fibered solid torus. After gluingand extending the foliation to all k tori, we get a manifold M and a collapsing map s : M → B where B is the surface of genus g and r boundary components.If a manifold M can be constructed like this, we say that it is a Seifert manifold and the map s : M → B is a Seifert fibering for M . We denote the resulting manifold after gluing k tori by M ( g, r, ( α , β ) , . . . , ( α k , β k )) . Each pair ( α i , β i ) is called Seifert pair and we say that it is normalized when ≤ β i < α i .We have, by definition and the discussion above, the following lemma and corollary. Lemma 3.2.
Let M → B be a Seifert fibering. If a fiber f has a neighborhood diffeomorphic to a ( p, q ) -solid torus, then the there exists b ∈ Z such that the (possibly unnormalized) Seifert invariantcorresponding to f is ( q, − b ) with bp ≡ q . Conversely, the special fiber f corresponding to aSeifert pair ( α, β ) has a neighborhood diffeomorphic as a circle bundle to a ( − c, α i ) -solid torus with cβ ≡ α . Corollary 3.3.
Let φ : Σ → Σ be an orientation preserving periodic automorphism of a surface Σ oforder n and let Σ φ be the corresponding mapping torus. Let x ∈ Σ be a point whose isotropy groupin the group < φ > has order k with n = k · s . Then φ s acts as a rotation in a disk around x withrotation number p/k for some p ∈ Z and the (possibly unnormalized) Seifert pair of M φ correspondingto the fiber passing through x is ( k, − b ) with bp ≡ k .Proof. That φ s acts as a rotation in a disk D ⊂ Σ around x with rotation number p/k for some p ∈ Z > follows from the fact that x is a fixed point for φ n/k . By construction of the mapping torusof Σ we observe that the two mapping tori M φ | D (cid:39) D φ n/k are diffeomorphic where D is a small diskaround x . By definition of fibered torus we have that D φ n/k (cid:39) T p,k . The rest follows from Lemma 3.2above. (cid:3) Plumbing graphs.
A plumbing graph is a decorated graph that encodes the information torecover the topology of a certain -manifold. As with Seifert manifolds, we fix notation and conven-tions.This is the decoration and its corresponding meaning: • Each vertex corresponds to a circle bundle. It is decorated with integers e i (placed on top)and g i placed on bottom. If a vertex has valency v i consider the circle bundle over the surfaceof genus g i and v i boundary components and pick a section on the boundary so that theglobal Euler number is e i . When g is omitted it is assymed to be . • Each edge tells us that the circle bundles corresponding to the ends of the edge are glued alonga boundary torus by the gluing map J ( x, y ) = ( y, x ) defined with respect to section × fiber on each boundary torus. • An and ending in an arrowhead represents that an open solid torus is removed from thecorresponding circle bundle from where the edge comes out.The construction of the -manifold associated to a plumbing graph is clear from the description ofits decoration above.We point out a minor correction to an argument in [Neu81] and reprove a known lemma which iscrucial in Section 5 (see discussion afterwards in Remark 3.7). Lemma 3.4.
Let Λ be a plumbing graph. If a portion of Λ has the following form: ... ... n − e − e − e k n . . . − e Then the piece corresponding to the node n is glued to the piece corresponding to the node n along a torus by the matrix G = (cid:0) a bc d (cid:1) with det( G ) = − and where − b/a = [ e , . . . , e k ] with the numbers in brackets being the continued fraction − ba = e − e − e − ... . If a portion of Λ has the following form: ... n − e − e − e k . . . − e Then the piece corresponding to the node n i is glued along a torus to the boundary of a solid torus D × S by the matrix (cid:0) a bc d (cid:1) with − d/c = [ e , e , . . . , e k ] .Proof. Let T := D × S be a solid torus naturally foliated by circles by its product structure. Let s be the closed curve ∂D × { } and let f be any fiber on the boundary of the solid torus. Orient themso that { [ s ] , [ f ] } is a positive basis of H ( ∂T ; Z ) . If T , T (cid:48) are two copies of the solid torus. Then M i is the S -bundle T (cid:116) E i T (cid:48) where E i : ∂T → ∂T (cid:48) is the matrix (cid:18) − e i (cid:19) used in the gluing along the boundaries. In particular [ f ] = [ f (cid:48) ] in H ( M i ; Z ) . The − in the upperleft part reflects the fact that s inherits different orientations from the two tori.We treat the case first. If M i is the piece corresponding to the node n i with i = 1 , we havethat the gluing from M to M is M (cid:116) J ( A × S (cid:116) E A × S ) (cid:116) J · · · (cid:116) J ( A × S (cid:116) E k A × S ) (cid:116) J M Where A × S is the trivial circle bundle over the annlus A := [1 / , × S . Let ( r, θ ) be polarcoordinates for A . The two tori forming the boundary of A × S are oriented as boundaries of A × S .Observe that the map r ((1 / , θ ) , η ) = ((1 , θ ) , η ) is orientation reversing.We define s = { S / } × { } and f = { (1 / , } × S and orient them so that the ordered basis { [ s ] , [ f ] } is a positive basis for H ( S / × S ; Z ) . We define similarly s (cid:48) = { S } × { } , f (cid:48) = { (1 , } × S and orient them so that { [ s (cid:48) ] . [ f (cid:48) ] } is a positive basis for H ( S × S ; Z ) . Then the homology classes [ r ( s )] and [ r ( f )] form a negative basis. In fact [ s (cid:48) ] = − [ r ( s )] and [ f ] = [ r ( f )] . This is the reason ofthe matrices (cid:0) − (cid:1) in the Equation (3.5) below.So the gluing matrix G from a torus in the boundary of M to a torus in the boundary of M isgiven by the following composition of matrices: G = ( ) (cid:0) − (cid:1) (cid:0) − e k (cid:1) (cid:0) − (cid:1) ( ) · · · ( ) (cid:0) − (cid:1) (cid:0) − e (cid:1) (cid:0) − (cid:1) ( )= ( ) (cid:0) − − e k (cid:1) ( ) · · · ( ) (cid:0) − − e (cid:1) ( )= ( ) (cid:0) − − e k (cid:1) · · · (cid:0) − − e (cid:1) (3.5) ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 11
Observe that each matrix in the definition of G has determinant − so det( G ) = − because thereis an odd number. Hence G inverts orientation on the boundary tori, preserving the orientation onthe global manifoldd. The result about the continued fraction follows easily by induction on k .Now we treat similarly the case . The gluing from a boundary torus from M to ∂D × S is M (cid:116) J ( A × S (cid:116) E A × S ) (cid:116) J · · · (cid:116) J ( A × S (cid:116) E k D × S ) . Hence, by a similar argument to the previous case, the matrix that defines the gluing is G = (cid:0) − e k (cid:1) (cid:0) − (cid:1) ( ) · · · ( ) (cid:0) − (cid:1) (cid:0) − e (cid:1) (cid:0) − (cid:1) ( )= (cid:0) − e k (cid:1) (cid:0) − (cid:1) ( ) (cid:0) − − e k − (cid:1) · · · (cid:0) − − e (cid:1) = (cid:0) − e k (cid:1) (cid:0) − − e k − (cid:1) · · · (cid:0) − − e (cid:1) (3.6)By the expression in the last line we see that all matrices involved but the one on the left, havedeterminant so we get det ( G ) = − . Again, by induction on k the result on the continued fractionfollows straight from the las line. (cid:3) Remark 3.7.
Note the differences of the Lemma above with Lemma 5.2 and the discussion before itin [Neu81]: there the author does not observe that in each piece A × S , the natural projection fromone boundary torus to the other is orientation reversing. So the matrices (cid:0) − (cid:1) are not taken intoaccount there.In a more extended manner. The problem is with the claim that the matrix C (in equation ( ∗ ) pg. of [Neu81]) is the gluing matrix. The equation above equation ( ∗ ) in that page, describes thegluing between the two boundary tori as a concatenated gluing of several pieces. In particular youglue a piece of the form A × S with another piece of the same form using the matrix H k and then youglue these pieces a long J - matrices. Then it is claimed that “since A × S is a collar” then the gluingmatrix (up to a sign) is JH k J · · · JH J . However, notice that each piece A × S has two boundarytori, and they inherit "opposite" orientations. More concretely, the natural radial projection fromone boundary torus to the other is orientation reversing. So even, if they are a collar (which theyare), they interfere somehow in the gluing. That is why we add the matrices (cid:0) − (cid:1) between each J and each H k matrix. 4. Horizontal surfaces in Seifert manifolds
In this section, we study and classify horizontal surfaces of Seifert fiberings up to isotopy. Theresults contained here are known. The exposition that we choose to do here is useful for Section 5.We recall that we are only considering Seifert manifolds that are orientable with orientable basespace and with non-empty boundary. Horizontal surfaces in a orientable Seifert manifold with ori-entable base space are always orientable (see for example Lemma 3.1 in [Zul01]). So by our assump-tions only orientable horizontal surfaces appear. Let F be any fiber of the Seifert fibering M → B . Definition 4.1.
Let H be a surface with non-empty boundary which is properly embedded in M i.e. H ∩ ∂M = ∂H . We say that H is a horizontal surface of M if it is transverse to all the fibers of M . Definition 4.2.
Let H ( M ) be the set of all horizontal surfaces of M , we define H ( M ) := H ( M ) / ∼ where two elements H , H ∈ H ( M ) are related H ∼ H if their inclusion maps are isotopic. Let n := lcm( α , . . . , α k ) . We consider the action of the subgroup of the unitary complex numbersgiven by the n -th roots of unity c n := { e πim/n } with m = 0 , . . . , n − on the fibers of M . Theelement e πim/n acts on a typical fiber by a rotation of πm/n radians and acts on a special fiberwith multiplicity α i by a rotation of πmα i /n radians.We quotient M by the action of this group and denote ˆ M = M/c n the resulting quotient space.By definition, the action of c n preserves the fibers and is effective. The manifold ˆ M is then a Seifertmanifold where we have killed the multiplicity of all the special fibers of M . Hence it is a locallytrivial S -fibration over B and since ∂B (cid:54) = ∅ , it is actually a trivial fibration so ˆ M is diffeomorphicto B × S .Let π : M → ˆ M be the quotient map induced by the action of c n . Observe that ˆ M , seen as aSeifert fibering with no special fibers, has the same base space as M because the action given by c n preserves fibers. In particular we have the following commutative diagram (4.3) M ˆ MB πs ˆ s Where s (resp. ˆ s ) is the projection map from the Seifert fibering M (resp. ˆ M ) onto its base space B . Definition 4.4.
Let H be a horizontal surface in M . We say that H is well embedded if it is invariantby the action of c n . A horizontal surface H defines a linear map H ( M ; Z ) → Z by considering its Poincare dual. If H intersects a generic fiber m times, then it intersects a special fiber with multiplicity α , m/α ∈ Z times. This is because a generic fiber covers that special fiber α times. Hence, by isotoping anyhorizontal fiber, we can always find well-embedded representatives ˆ H ∈ [ H ] . Remark 4.5.
Observe that if H and and H (cid:48) are two well-embedded surfaces with [ H ] = [ H (cid:48) ] , then wecan always find a fiber-preserving isotopy h that takes the inclusion i : H (cid:44) → M to an homeomorphism h ( · ,
1) : H → H (cid:48) such that h ( H, t ) is a well-embedded surface for all t . This fact help us prove thefollowing: Lemma 4.6.
There is a bijection π (cid:93) : H ( M ) → H ( ˆ M ) induced by π .Proof. Let [ H ] ∈ H ( M ) and suppose that H ∈ [ H ] is a well-embedded representative. Then clearly π ( H ) ∈ H ( ˆ M ) . If H (cid:48) is another well-embedded representative of the same class, then by Remark 4.5we have that [ π ( H )] = [ π ( H (cid:48) )] in H ( ˆ M ) . Hence the map π (cid:93) ([ H ]) := [ π ( H )] is well defined.The map π (cid:93) is clearly surjective because π − ( ˆ H ) is a well-embedded surface for any horizontalsurface ˆ H ∈ H ( ˆ M ) and hence π (cid:93) ([ π − ( ˆ H )]) = [ ˆ H ] .Now we prove that the natural candidate for inverse π − (cid:93) ([ ˆ H ]) := [ π − ( ˆ H )] is well-defined. Let [ ˆ H ] ∈ H ( ˆ M ) with ˆ H ∈ [ ˆ H ] a representative of the class. Let H := π − ( ˆ H ) . If [ ˆ H ] = [ ˆ H (cid:48) ] for some ˆ H (cid:48) in H ( ˆ M ) then [ π − ( ˆ H (cid:48) )] = [ π − ( ˆ H )] by just pulling back the isotopy between ˆ H and ˆ H (cid:48) to M bythe map π . Hence the map is well defined. By construction, it is clear that for any H ∈ H ( M ) wehave that π − (cid:93) ( π (cid:93) ([ H ]) = [ H ] so we are done. (cid:3) The objective of this section is to study H ( M ) but because of Lemma 4.6 above, it suffices to study H ( ˆ M ) .Fix a trivialization ˆ M (cid:39) B × S once and for all. We observe that since ∂B (cid:54) = ∅ , the surface B ishomotopically equivalent to a wedge of µ = 2 g + r − circles, denote this wedge by ˜ B . Observe that H ( M ) is in bijection with multisections of ˜ B × S → ˜ B up to isotopy. Multisections are multivaluedcontinuous maps from ˜ B to ˜ B × S . Lemma 4.7.
The elements in H ( ˜ B × S ) are in bijection with elements of H ( ˜ B × S ; Z ) = H ( ˜ B ; Z ) ⊕ Z that are not in H ( ˜ B ; Z ) ⊕ { } . Oriented horizontal surfaces that intersect positively any fiber of ˜ B × S are in bijection with elements of H ( ˜ B ; Z ) ⊕ Z > . Proof.
We have that H ( B ; Z ) ⊕ Z = Z µ ⊕ Z . Given an element ( p , . . . , p µ , q ) = k (( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) )) ∈ Z µ × Z with q (cid:54) = 0 and ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) irreducible (seeing Z µ × Z as a Z -module). Let p (cid:48) j q (cid:48) = k j p (cid:48)(cid:48) j k j q (cid:48)(cid:48) with p (cid:48)(cid:48) j /q (cid:48)(cid:48) an irreducible fraction. Consider in each S k × S , k j disjoint copies of the closed curveof slope p (cid:48)(cid:48) j /q (cid:48)(cid:48) . We denote the union of these k j copies by ˜ H j . We observe that ˜ H j intersects C in k j q (cid:48)(cid:48) = q (cid:48) points for each j . We can, therefore, isotope the connected components of each ˜ H j so that (cid:83) j ˜ H j intersects C in just q (cid:48) points. We do so and consider the set (cid:83) j ˜ H j . The horizontal surface ˜ H of ˜ B × S associated to k (( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) is k disjoint parallel copies of (cid:83) j ˜ H j .On the other direction, given an element [ H ] ∈ H ( ˜ B × S ) . Let [ H ] denote also the class of anyhorizontal surface in H ( ˜ B × S ; Z ) and we simply have q = [ H ] · C . And we have p i = [ H ] · [ S i × { } ] .That is, the corresponding element in H ( B ; Z ) ⊕ Z is the Poincaré dual of the class of H in thehomology of ˜ B × S . (cid:3) ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 13
Lemma 4.8. ˜ H is connected if and only if the element ( p , . . . , p µ , q ) is irreducible in H ( ˆ M ; Z ) (cid:39) H ( ˜ B ; Z ) ⊕ Z .Proof. We know that by construction ˜ H ∩ C are q points. It is enough to show that these q pointslie in the same connected component since any other part of ˜ H intersects some of these points. Welabel the points cyclically according to the orientation of C . So we have c , . . . , c q ∈ C . We recallthat S j × S ∩ ˜ H is formed by k j parallel copies of the closed curve of slope p (cid:48) j /q (cid:48) with k j p (cid:48) j k j q (cid:48) = p j q .Hence the point x i is connected by these curves with the points c i + tk j mod q . Since ( p , . . . , p µ , q ) isirreducible then gcd( p , . . . , p µ , q ) = 1 and hence gcd( k , . . . , k µ ) = 1 . Therefore the equation i + t k + · · · + t µ k µ = j mod q admits an integer solution on the variables t , . . . , t µ for any two i, j ∈ { , . . . , q } . This proves thatthe points c i and c j are in the same connected component in ˜ H .Conversely if the element is not irreducible, then ( p , . . . , p µ , q ) = k ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) for ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) irreducible and k > . Then, by construction, ˜ H is formed by k disjoint copies of the connected hor-izontal surface associated to ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) (cid:3) Handy model of a Seifert fibering.
We describe a particularly handy model of the Seifert fiberingthat we use in Section 5. The idea is taken from a construction in [Hat07]. For each i = 1 , . . . , k let x i ∈ B be the image by s : M → B of the special fiber F i . We pick one boundary component of thebase space and denote it by L . For each i = 1 , . . . , k pick an arc l i properly embedded in B and withthe end points in L (i.e. with l i ∩ L = ∂l i ) in such a way that cutting along l i cuts off a disk D i thatcontains x i and no other point from { x , . . . , x k } . We pick a collection of such arcs l , . . . , l k pairwisedisjoint. We define B (cid:48) := B \ (cid:71) i int ( D i ) where int ( · ) denotes the interior. See Figure 4.1 below and observe that B and B (cid:48) are diffeomorphic. x x x D D D Figure 4.1.
We see the base space B of a Seifert manifold. It has genus and boundarycomponents. The points are the image of the special fibers by the projection s and if we cut alongthe three red arcs, we get the surface B (cid:48) . Let M (cid:48) := s − ( B (cid:48) ) . Since M (cid:48) contains no special fibers and ∂B (cid:48) (cid:54) = ∅ then M (cid:48) is diffeomorphic asa circle bundle to B (cid:48) × S . Recall that s − ( D i ) is a solid torus of type ( p i , α i ) with p i β i ≡ α i (see Lemma 3.2).Summarizing, the handy model consists of:i) A system of arcs l , . . . , l k as explained.ii) A trivialization of M (cid:48) , that is an identification of M (cid:48) with B (cid:48) × S .iii) Identifications of s − ( D i ) with the corresponding model T p i ,α i for each i = 1 , . . . , k . Remark 4.9.
Let A i be the vertical annulus s − ( l i ) . A properly embedded horizontal disk D ⊂ s − ( D i ) intersects A i in α i disjoint arcs by definition of the number α i . Since a horizontal surface H intersects each typical fiber the same number of times we get that H must meet each fiber t · lcm( α , . . . , α k ) = t · n times for some t ∈ Z > . If a horizontal surface meets t · n times a typical fiber,then it meets t · n/α i times the special fiber F i . Lemma 4.10.
There is a bijection between H ( M ) and HS ( M ) := { γ ∈ H ( M ; Z ) : γ ([ C ]) (cid:54) = 0 } where C is a generic fiber of M .Proof. Clearly, an element [ H ] ∈ H ( M ) can be seen as the dual of a -form γ with γ H ( C ) (cid:54) = 0 .To see that there is a bijection, take a handy model for M (we use notation described there).Then we observe that given a γ ∈ HS ( M ) , it restricts to a -form in H ( M (cid:48) ; Z ) . The manifold M (cid:48) isdiffeomorphic to a product, so by Lemma 4.7, there is a horizontal surface in H ( M (cid:48) ) representing therestriction of γ to M (cid:48) . It also restricts as a -form in H ( s − ( B ); Z ) where we recall that s − ( B ) is adisjoint union of tori, each containing a special fiber of M . If γ ([ C ]) = n then, γ ([ F i ]) = n/α i ∈ Z sowe can see the dual of γ | s − ( B ) as an union of n/α i disks in each of the tori s − ( D i ) for all i . Each ofthese disks intersects α i times the annulus s − ( l i ) . So we can glue the horizontal surface representedby γ | M (cid:48) with these disks to produce a horizontal surface in all M . By construction, this horizontalsurface represents the given γ ∈ H ( M ; Z ) . (cid:3) Lemma 4.11.
Let ˆ H ∈ H ( ˆ M ) and H := π − ( ˆ H ) . Then H is connected if and only if ˆ H is connected.Proof. If H is connected, then so is ˆ H because π is a continuous map.Suppose now that ˆ H is connected. If π − ( ˆ H ) is not connected, then it is formed by parallel copiesof diffeomorphic horizontal surfaces. Each of them is sent by π onto ˆ H and each of them representsthe same element in HS ( M ) . But, by Lemma 4.10 HS ( M ) is in bijection with H ( M ) which, byLemma 4.6, is in bijection with H ( ˆ M ) . So we get to a contradiction. (cid:3) By construction, we have established the correspondences(4.12) HS ( M ) ←→ H ( M ) ←→ H ( ˆ M ) ←→ H ( B ; Z ) ⊕ Z \ H ( B ; Z ) ⊕ { } Where the first correspondence is Lemma 4.10, the second is Lemma 4.6 and the last one isLemma 4.7.Actually if we fix an orientation on the manifold and the fibers and we restrict ourselves to orientedhorizontal surfaces that intersect positively the fibers of M , these are parametrized by elements in H ( B ; Z ) ⊕ Z > . From now on we restrict ourselves to oriented horizontal surfaces H with H · C > ,that is, those whose oriented intersection product with any typical fiber is positive. Also the fibersare assumed to be oriented. This orientation induces a monodromy on each horizontal surface. Remark 4.13.
Let Σ be a surface with boundary and φ : Σ → Σ a periodic automorphism and let Σ φ be the corresponding mapping torus which is a Seifert manifold. The manifold Σ φ fibers over S and we can see Σ as a horizontal surface of Σ φ by considering any of the fibers of f : Σ φ → S .Now let Σ φ be the orbit space of Σ which is also the base space of Σ φ . Let m be the lcm of themultiplicities of the special fibers of the Seifert fibering and let Σ φ /c m be the quotient space resultingfrom the action of c m on Σ φ . We observe, as before, that Σ φ /c m is diffeomorphic to Σ φ × S butthere is not preferred diffeomorphism between them. A trivialization is given by a choice of a sectionof Σ φ /c m → Σ φ .Let [ S ] , . . . , [ S µ ] be a basis of the homology group H (Σ φ ; Z ) where each S i is a simple closedcurve in Σ φ . Let C be any fiber of of Σ φ /c m . Let w, ˆ w : Σ φ → Σ φ /c m be two sections, then we havetwo different basis of the homology of H (Σ φ /c m ; Z ) induced by these two sections. For instance { [ w ( S )] , . . . , [ w ( S µ )] , [ C ] } and { [ ˆ w ( S )] , . . . , [ ˆ w ( S µ )] , [ C ] } . Let
Σ = f − (0) be the horizontal surface that we are studying and let ˆΣ := π (Σ) where π is thequotient map Σ φ → Σ φ /c m . Then ˆΣ is represented with respect to the (duals of the) two basis byintegers ( p , . . . , p µ , q ) and (ˆ p , . . . , ˆ p µ , q ) respectively and p i ≡ ˆ p i mod q for all i = 1 , . . . , µ becausea section differs from another section in a integer sum of fibers at the level of homology.So the numbers p , . . . , p µ are well defined modulo Z q regardless of the trivialization chosen for Σ φ /c m . Also by the discussion above, we see that if we fix a basis of H ( B ; Z ) , then all the elementsof the form ( p + n q, . . . , p µ + n µ q, q ) represent diffeomorphic horizontal surfaces with the samemonodromy. That there exists an diffeomorphism of M preserving the fibers that sends H to H comes from the fact that on a torus S × S , there exist a diffeomorphism preserving the verticalfibers { t } × S that sends the curve of type ( p, q ) to the curve of type ( p + kq, q ) for any k ∈ Z : the k -th power of a left handed Dehn twist along some fiber { pt } × S that is different from C . ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 15 Translation Algorithms
Every mapping torus arising from a tête-à-tête graph is a Seifert manifold so it admits a (star-shaped) plumbing graph. The monodromies induced on horizontal surfaces of Seifert manifolds areperiodic.In this section we describe an algorithm that, given a general tête-à-tête graph, produces a star-shaped plumbing graph together with the element in cohomology modulo Z q corresponding to thehorizontal surface given by the tête-à-tête graph. We also describe the algorithm that goes in theopposite direction.5.1. From general tête-à-tête graph to star-shaped plumbing graph.
We first state a knownproposition that used in the algorithm. It can be found in several references in the literature. See forexample [NR78] or [Ped09] .
Proposition 5.1.
Let M ( g, r ; (ˆ α , ˆ β ) , . . . , (ˆ α k , ˆ β k )) be a Seifert fibering. Then it is diffeomorphicas a circle bundle to a Seifert fibering of the form M ( g, r ; (1 , b ) , ( α , β ) , . . . , ( α k , β k )) where ≤ β i < α i for all i = 1 , . . . , k . If the surface admits a horizontal surface, then b = − (cid:80) i β i α i .The corresponding plumbing graph associated to the Seifert manifold is . . .. . .. . . ... b − b − b − b k − − b k − b M − − b M − b k − M k − − b kM k g ... r Figure 5.1.
Plumbing graph for a Seifert manifold where the numbers b ij are the continuous fraction expansion for α i /β i , that is α i β i = b i − b i − b i − ... Let (Γ , P , σ ) be a general tête-à-tête structure. For simplicity, we suppose that Γ is connected.This includes as particular cases pure tête-à-tête graphs and relative tête-à-tête graphs. Let φ Γ bea truly periodic representative of the tête-à-tête automorphism and let Σ φ Γ be the mapping torus ofthe the diffeomorphism φ Γ : Σ → Σ . The mapping torus given by a periodic diffeomorphism of asurface is a Seifert manifold. We describe an algorithm that takes (Γ , P , σ ) as input and returns asoutput:(1) The invariants of a Seifert manifold: M ( g, r ; (1 , b ) , ( α , β ) , . . . , ( α k , β k )) diffeomorphic to the mapping torus Σ φ Γ . It is represented by a star-shaped plumbing graph Λ corresponding to the Seifert manifold and(2) a tuple ( p , . . . , p µ , q ) with p i ∈ Z /q Z and q ∈ Z > representing the horizontal surface givenby Σ with respect to some basis of the homology H ( B ; Z ) (cid:39) Z µ of the base space B of M . Step 1.
We consider Γ φ Γ , that is the quotient space Γ / ∼ where ∼ is the equivalence relationinduced by the action of the safe walks on the graph. This graph is nothing but the image of Γ bythe projection of the branched cover p : Σ → Σ φ Γ onto the orbit space.The map p | Γ : Γ → Γ φ Γ induces a ribbon graph structure on Γ φ Γ . We can easily get the genus g and number of boundary components r of the thickening of Γ φ Γ from the combinatorics of the graph.This gives us the first two invariants of the Seifert manifold. Step 2.
Let sv (Γ) be the set of points with non trivial isotropy subgroup in < φ Γ > . This is theset of branch points of p : Σ → Σ φ Γ by definition.Let v ∈ sv (Γ) . Then there exists m < n with n = m · s such that v is a fixed point of φ m Γ (take m the smallest natural number satsifying that property). We can therefore use Corollary 3.3. We getthat φ m Γ acts as rotation with rotation number p/s in a small disk centered at v . So around the fiber corresponding to the vertex v , the Seifert manifold is diffeomorphic fiberwise to a p, s -torus. and thecorresponding Seifert pair ( α v , β v ) is given by ( α v , β v ) = ( s, − b ) with bp ≡ q .We do this for every vertex in sv (Γ) and we get all the Seifert pairs. Step 3.
Since we have already found a complete set of Seifert invariants, we have that M ( g, r ; ( α , β ) , . . . , ( α k , β k ) , (˜ α k +1 , ˜ β k +1 )) is diffeomorphic to the mapping torus of the pair (Σ , φ Γ ) .Now we use Proposition 5.1 to get the normalized form of the plumbing graph associated to theSeifert manifold. Step 4.
For this step, recall Section 4 and notation introduced there. Fix a basis [ S ] , . . . , [ S µ ] of H ( B ; Z ) where S i is a simple closed curve contained in Γ φ Γ .Let m := lcm( α , . . . , α k ) . Observe that necessarily m | n so n = m · q (with n the order of φ Γ ).This number q that we have found is the last term of the cohomology element we are looking for.Of course, it is also the oriented intersection number of ˆΓ = π (Γ) with C . (recall π was the projection M → M/c m =: ˆ M ).Pick any basis of H ( B ; Z ) by picking a collection of circles S , . . . , S k contained in the orbit graph Γ φ Γ ⊂ B that generate the homology of the graph.Now if we intersect the graph Γ /c m ∩ S i × C with the torus over one of the representatives ofthe basis, we get a collection of k i closed curves, each one isotopic to the curve of slope p (cid:48) i /q (cid:48) i where q (cid:48) i · k i = q and p i = p (cid:48) i · k i = p i . This number, p i , is the i − th coordinate of the cohomology elementwith respect to the fixed basis.We can compute p , . . . , p k directly. Let S i be one of the generators of H (Γ φ Γ ; Z ) . Let ˆ S i :=Γ /c m ∩ S i × S where S i × S ⊂ Γ φ Γ × S ; and let ˜ S i := p | − ( ˆ S i ) . Observe that ˆ S i consists of k i disjoint circles and that k i divides q . Let q (cid:48) i = q/k i .Pick a point z ∈ S i which is not in the image by π of a special fiber. Then ˆ π − ( z ) ∩ Γ /c m consistsof q points lying in the k i connected components of ˆ S i . Pick one of these connected componentsand enumerate the corresponding q (cid:48) i points in it using the orientation induced on that connectedcomponent by the given orientation of S i . Then we have the points z , . . . , z q (cid:48) i . We observe thatby construction, these points lie on the same fiber in ˆ M and this fiber is oriented. Follow the fiberfrom z in the direction indicated by the orientation, the next point is z t i , with t i ∈ { , . . . , q (cid:48) i } . Wetherefore find that this connected component of ˆ S i lies in S i × S as the curve with slope ( t i − /q (cid:48) i and so p i = ( t i − · k i .5.2. From star-shaped plumbing graph to tête-à-tête graphs.
The input that we have is:i) A Seifert fibering of a manifold M .ii) A horizontal surface given by an element in H ( B × S ; Z ) that does not vanish on a typicalSeifert fiber.The output is:(1) A general, relative or pure tête-à-tête graph such the induced mapping toru is diffeomorphicto the given plumbing manifold in the input. And such that the thickening of the graph,represents the horizontal surface given. Step 1.
We start with a Seifert fibering M ( g, r ; ( α , β , . . . , α k , β k ) .We fix a model of the Seifert fibering as in Handy model of a Seifert fibering. We recall that themodel consists of the following data:i) The Seifert fibering s : M → B where B is a surface of genus g and r boundary components.ii) A collection of arcs { l i } with i = 1 , . . . , k properly embedded in B where the boundary ofthese arcs lie in one chosen boundary component of B . These satisfy that when we cut alongone of them, say l i we cut off a disk denoted by D i from B that contains the image of exactlyone special fiber, we denote the image of this fiber by x i .iii) We have an identification of each solid torus s − ( D i ) with the corresponding fibered solidtorus T p i ,q i with q i = α i and − p i β i ≡ α i and < p i < q i . Step 2.
Observe that B is homotopic to a wedge of µ = 2 g − r + 1 circles that does not containany x i for i = 1 , . . . , i . We can see this wedge as a spine embedded in B . Denote by c the commonpoint of all the circles. Now we embed disjoint segments e i with i = 1 , . . . , k where each one satisfiesthat one of its ends lies in the spine and the other end lies in x i . Also, they do not intersect the ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 17 wedge of circles at any other point and they also e i does not intersect any D j for j (cid:54) = i . We denotethe union of the wedge and these segments by ˜Λ and observe that ˜Λ is a spine of B . Step 3.
We suppose that the element in H ( ˆ M ; Z ) given is irreducible, otherwise if it is of the form k ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) with ( p (cid:48) , . . . , p (cid:48) µ , q (cid:48) ) irreducible, we take the irreducible part, carry out the followingconstruction of the corresponding horizontal surface and then take k parallel copies of this surface.Recall Equation (4.3) for the definition of the maps s, ˆ s and π .Once and for all, fix a trivialization ˆ M (cid:39) B × S . We assume that the element ( p , . . . , p µ , q ) ∈ H ( ˆ M ; Z ) is expressed with respect to the dual basis [ S ] , . . . , [ S µ ] , [ C ] where the first µ are circles ofthe wedge embedded in B and [ C ] is the homology class of C := ˆ s − ( c ) .For each i = 1 , . . . , µ , consider the torus ˆ s − ( S i ) which is naturally trivialized by the trivializationof ˆ M . We pick in it k i copies of the curve of slope p (cid:48) i /q (cid:48) where p i /q = k i p (cid:48) i /k i q (cid:48) . For each i , the curvesconstructed this way in ˆ s − ( S i ) intersect q times the curve C . Hence we can isotope them so thatall of them intersect C in the same q points. We denote the union of these curves by ˆΛ (cid:48) . We assumethat ˆΛ (cid:48) projects to ˜Λ \ (cid:83) e i by B × S → B .By construction, ˆΛ (cid:48) is a ribbon graph for the surface horizontal surface ˆ H ⊂ ˆ M . Observe that s (ˆΛ (cid:48) ) (cid:54) = ˜Λ . However s (ˆΛ (cid:48) ) is also a spine of B (it coincides with the wedge of circles in B ).Define Λ (cid:48) := π − (ˆΛ (cid:48) ) . By the definition of π , this graph can also be constructed by taking in eachof the tori π − (ˆ s − ( S i )) = s − ( S i ) , k i copies of the curve of slope p (cid:48) i /n . Which by construction allintersect in n points in s − ( c ) . Step 4.
Now we describe π − (ˆ s | − H ( e i )) for each i = 1 , . . . , k . First we observe that it is equal to s | − H ( e i ) which is a collection of q · n/α i disjoint start shaped graphs. Each star-shaped piece has α i .To find out the gluings of these arms with Λ (cid:48) one looks as the structure of s − ( D i ) as a ( c, α i ) - solidtorus. To visualize it, place the q · n/α i star-shaped pieces in a solid cylinder D × [0 , and identifytop with botton by a c/α i ) rotation. The fibers of the fibered torus give the monodromy on the endof the arms and the attaching to Λ (cid:48) .We define Λ as the union of Λ (cid:48) with these star-shaped pieces Step 5.
The embedding of H in the Seifert manifold defines a diffeomorphism φ : H → H in thefollowing way. Let x ∈ H and follow the only fiber of the Seifert manifold that passes through x inthe direction indicated by its orientation, we define φ ( x ) as the next point of intersection of that fiberwith H .To describe φ up to isotopy it is enough to give the rotation numbers of φ around each boundarycomponent of H plus some spine invariant by φ . By construction Λ is an invariant graph. The fibersof the Seifert fibering give us an automorphism on the graph Λ . To get the rotation numbers, we cutthe thickening H along Λ and we get a collection of cylinders Λ j × [0 , with j = 1 , . . . , r (cid:48) .Now we invoke [FdBPPPC17, Theorem 5.12] if the monodromy leaves at least boundary compo-nent invariant and we invoke Theorem 2.6 if the monodromy does not leave any boundary componentinvariant. This gives us a constructive method to find a graph (which in general will be differentfrom (cid:83) µi =1 S i (cid:83) kj =1 e j ) containing all branch points in B such that it is a retract of B and such thatit admits a metric that makes its preimage a tête-à-tête graph.6. Examples
We apply the algorithms developed in the previous sections to two examples.
Example 6.1.
Suppose we are given the bipartite complete graph Γ of type , with the cyclicordering induced by placing and vertices in two horizontal parallel lines in the plane and takingthe joint of the two sets in that plane. Give each edge length π/ . This metric makes it into a tête-à-tête graph as we already know. Let φ Γ be a periodic representative of the mapping class inducedby the tête-à-tête structure.Let’s find the associated invariants. One can easily check that the orbit graph is just a segmentjoining the only two branch points so the orbit surface is a disk and hence g = 0 and r = 1 .The map p : Σ → Σ φ Γ has two branch points that correspond to two Seifert pairs. Let r be thebranching point in which preimage lie the points of valency . We choose any of those pointsand denote it p , now φ acts as a rotation with rotation number / in a small disk around p . Hence, the associated normalized Seifert pair is (11 , . Note that · ≡ − and that < < . Equivalently for the other point we find that φ is a rotation with rotation number π/ φ φ Figure 6.1.
On the left we see the tête-à-tête graph K , . On the right we see a small neigh-bourhood of a vertex of valency where φ acts as the rotation r / radians. Equivalently, for avertex of valency , we see that φ acts as the rotation r / . / when restricted to a disk around any of the vertices of valency . Hence, the correspondingnormalized Seifert pair is (4 , .Computing the continued fraction we have that = [2 , , , and = [4] . For computing thenumber b we think of the surface resulting from extending the periodic automorphism to a diskcapping off the only boundary component of Σ . By a similar argument, since the rotation numberinduced on the boundary is − / , this would lead to a new Seifert pair (44 , . Since these arenormalized Seifert invariants, the new manifold is closed and admits a horizontal surface, we can useProposition 5.1 and compute the number b as − / − / − /
44 = − .So the plumbing diagram corresponding to the mapping torus of Σ by φ Γ is the following. − − − − − − − which, up to contracting the bamboo that ends in the arrowhead, coincides with the dual graphof the resolution of the singularity of x + y at .Finally, we are going to compute the element that the surface Σ represents in the homology group H (Σ φ Γ ) ⊕ Z . First observe that since Σ φ Γ is a disk, the group is isomorphic to ⊕ Z . This tells usthat the only possible choices of multisections in the bundle Σ φ Γ × S are classified (up to isotopy) bythe elements (0 , k ) with k (cid:54) = 0 . The element (0 , k ) corresponds to k parallel copies of the disk Σ φ Γ .In our case, there is only one such disk so the element is (0 , . Example 6.2.
Suppose we are given the following plumbing graph: − − − − Figure 6.2
We are indicated two of the invariants of the Seifert manifold: the genus of the base space g = 0 and its number of boundary components r = 2 . The base space B is therefore an annulus.We compute the Seifert invariants by interpreting the weights on the two bamboos of the plumbinggraph as numbers describing continued fractions. We get [2 ,
2] = 3 / and [2] = 2 so the Seifert pairsare (3 , and (2 , . So the corresponding Seifert fibering s : M → B has two special fibers F ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 19 (for the pair (2 , and F (for the pair (3 , ). Using Lemma 3.2 we have that the Seifert fibercorresponding to the pair (3 , has a tubular neighborhood diffeomorphic to the fibered solid torus T , ; this is because − · ≡ . Analogously, the fiber corresponding to the Seifert pair (2 , has a tubular neighborhood diffeomorphic to the fibered solid torus T , .Now we fix a model for our Seifert manifold. Take an annulus as in Figure 6.3. Now we usethe kind of model explained in Figure 4.1; we choose a boundary component and we pick properlyembedded arcs (with their boundaries lying on the chosen boundary component) in such a way thatcutting along one of them cuts off a disk containing only one of the two images by s of the specialfibers; over those disks in M lie the two corresponding fibered solid tori. Let d be the point lyingunder π ( F ) and let a be the point lying under the fiber π ( F ) . We pick an embedded graph which isa spine of B as in Figure 6.3 below, that is, the graph is the union of: a circle whose class generatesthe homology of the base space. We denote it by S ; a segment joining a point c ∈ S with the vertex d . We denote this segment by D and a segment joining a point b ∈ S with the vertex a . We denotethis segment by A . See Figure 6.3.We denote this graph by ˜Λ . abcd D A Figure 6.3.
This is the base space B of the Seifert fibering. In red we see ˜Γ which is formed by acircle and two segments attached to it that end at the image by s of the special fibers. The dashedlines represents the properly embedded arcs Now we consider ˆ M which is diffeomorphic to B × S which is homotopically equivalent to ˜Γ × S .We denote the projection on B by ˆ s : ˆ M → B . The map π : M → ˆ M satisfies that ˆ s ◦ π = s .The piece of information missing from the input is the horizontal surface. Suppose we are giventhe element (1 , ∈ H ( B ; Z ) ⊕ Z with respect to the basis formed by the class of S . Then, theintersection of the horizontal surface ˆ H ⊂ ˆ M with the torus ˆ S := ˆ s − ( S ) is a curve of slope / . Wealso have that ˆ s − ( A ) consists of two segments, as well as ˆ s − ( D ) . See figure Figure 6.4. Figure 6.4.
This is ˆ M together with the base space under it. Lying over the graph ˜Γ we can seethe graph ˆ G whose thickening is the horizontal surface ˆ H (the blue helicoidal ramp on the figure).Also we see that lying over the circle of ˜Γ lies the closed curve in ˆΓ that is a curve of slope / inthe torus ˆ s − ( S ) . The horizontal surface that we are looking for is H := π − ( ˆ H ) that is the thickening of π − (ˆΓ) .To know the topology of H and the action on it of the monodromy, we construct the ribbon graph π − (ˆΓ) . We observe that lcm(2 ,
3) = 6 so π − ( ˆ S ) is the curve of slope / on the torus s − ( S ) . We also have that s − ( a ) = ( π ◦ ˆ s ) − ( a ) consists of and s − ( A ) consists of segments separated ingroups giving valency to each of the points in points. Equivalently s − ( d ) consists of vertices and s − ( D ) of segments naturally separated by pairs. The fact that s − ( S ) is a curve of slope / ,give us the combinatorics of the graph. Using notation of 6.5, and the rotation numbers associatedto each of the two Seifert pairs, we have that the graph is that of Figure 6.5. a a a a a a a a a a a a b b b b b b b b b b b b c c c c c c c c c c c c d d d d d d d d d d d d xφ ( x ) yφ ( y ) Figure 6.5.
The graph π − (ˆΛ) in black. The letters with subindexes are interpreted like this:the vertex a i is glued to the vertex b i and the vertex d i is glued to the vertex d i . In red wesee a path from x to φ ( x ) used to compute the rotation number of φ with respect to the outerboundary component; we observe that the outer boundary component retracts to edges (eachedge is counted twice if the boundary component retracts to both sides of the edge), and the redpath covers of these edges. You can easily compute from the ribbon graph that the surface has boundary components andgenus . Since it has only two boundary components, each of them is invariant by the action ofthe monodromy induced by the orientation on the fibers. We compute their rotation numbers asexplained on Step of the algorithms. We observe that the ”outer” boundary component retracts to edges where an edge is counted twice if the boundary component retracts to both sides of it. Wepick a point x and observe that the monodromy indicated by the orientation of the fibers takes it tothe point inmediately above it φ ( x ) . Now we consider a path ”turning right” starting at x and observethat it goes along edges before reaching φ ( x ) . Hence, the rotation number of φ with respect to thisboundary component is / . Similarly, we observe that the other boundary component retractsto edges and by a similar procedure we can check that φ also has a rotation number / withrespect to this other boundary component. See figure Figure 6.5.Following the construction in [FdBPPPC17, Theorem 5.12], we should put a metric on ˆ G so thatthe part where the outer boundary component retracts has a length of π/ and the same for theother boundary component. But this is impossible given the combinatorics of the graph. That meansthat this graph does not accept a tête-à-tête metric. However theorem [FdBPPPC17, Theorem 5.12]gives us a procedure to find a graphadmittin a tête-à-tête metric producing the given monodromy.In this case, it is enough to consider the following graph. ENERAL TÊTE-À-TÊTE GRAPHS AND SEIFERT MANIFOLDS 21
Figure 6.6.
Graph ˜Γ that admits a tête-à-tête metric. If we call this graph ˜Γ we see that that Γ := p − (˜Γ) is the following graph: a a a a a a a a a a a a ˆ a ˆ a ˆ a a ˆ a ˆ a ˆ a ˆ a ˆ a ˆ a ˆ a ˆ a b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b ˆ b b b b b b b b b b b b c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c ˆ c c c c c c c c c c c c d d d d d d d d d d d d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d ˆ d Figure 6.7.
The tête-à-tête graph Γ . The notation means that a i is glued to b i , ˆ a i to ˆ b i , c i to d i and ˆ c i to ˆ d i for all i = 1 , . . . , . By setting each of the two edges of the circle ˜Γ has length π/ , then Γ is a pure tête-à-tête graphmodelling the action of the monodromy on the horizontal surface H . References [A’C10] Norbert A’Campo. Tête-à-tête twists and geometric monodromy. Preprint, 2010.[EN85] David Eisenbud and Walter D. Neumann.
Three-dimensional link theory and invariants of plane curvesingularities. , volume 110 of
Annals of Mathematics Studies . Princeton University Press, 1985.[FdBPPPC17] Javier Fernandez de Bobadilla, María Pe Pereira, and Pablo Portilla Cuadrado. Representations ofsurface homeomorphisms by tête-à-tête graphs. June 2017.[Gra14] Christian Graf. Tête-à-tête graphs and twists. Aug 2014.[Gra15] Christian Graf. Tête-à-tête twists: Periodic mapping classes as graphs. Jul 2015.[Hat07] Allen Hatcher. Notes on basic -manifold topology, 2007.[HNK71] F. Hirzebruch, W. D. Neumann, and S. S. Koh. Differentiable manifolds and quadratic forms . Mar-cel Dekker, Inc., New York, 1971. Appendix II by W. Scharlau, Lecture Notes in Pure and AppliedMathematics, Vol. 4.[JN83] Mark Jankins and Walter D. Neumann.
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