Splice Diagram Singularities and The Universal Abelian Cover of Graph Orbifolds
aa r X i v : . [ m a t h . G T ] N ov SPLICE DIAGRAM SINGULARITIES AND THE UNIVERSALABELIAN COVER OF GRAPH ORBIFOLDS
HELGE MØLLER PEDERSEN
Abstract.
Given a rational homology sphere M , whose splice diagram Γ( M ) satisfy the semigroup condition, Neumann and Wahl were able to define acomplete intersection surface singularity called splice diagram singularity from Γ( M ) . They were also able to show that under an additional hypothesis on M called the congruence condition, the link of the splice diagram singularity isthe universal abelian cover of M . In this article we generalize the congruencecondition to the class of orbifolds called graph orbifold. We show that undera small additional hypothesis, this orbifold congruence condition implies thatthe link or the splice diagram equations is the universal abelian cover. We alsoshow that any two node splice diagram satisfying the semigroup condition,is the splice diagram of a graph orbifold satisfying the orbifold congruencecondition. Introduction
The topology of an isolated complex surface singularity is determined by its link,which all turn to be among the class of -manifolds called graph manifolds, thatis the manifolds which only have Seifert fibered pieces in their JSJ-decomposition.There are several graph invariants of graph manifolds which is used to study them.The first is the plumbing diagram of a plumbed -manifold X such that our graphmanifold M is the boundary of X , this does give a complete invariant if one as-sumes the plumbing diagram is in a normal form (see [Neu81]), of which thereare several different. Now the plumbing diagrams can be quite large and does notalways display the properties of the manifold clearly, so we are interested in another invariant called splice diagram. Splice diagrams where original introduced in[EN85] and [Sie80] for integer homology sphere graph manifolds. It was then latergeneralized by Neumann and Wahl to rational homology spheres in [NW02]. Theyused it extensively in [NW05b] and [NW05a], and especially their use in [NW05a]is of interest to us.In [NW05a] they use the splice diagram of a singularity link M satisfying whatthey call the semigroup condition, to construct a set of equations called splice di-agram equations defining an isolated complete intersection surface singularity X .They then showed that if M satisfy an additional hypothesis called the congruencecondition, the link of X is the universal abelian cover of M . In [Ped10a] I showedthat the splice diagram of any graph manifold M always determines the universalabelian cover of M . So combining these to result one gets a nice description of theuniversal abelian cover of a graph manifold M , as the link of complete intersection,provided that there is some graph manifold M ′ with the same splice diagram satis-fying the congruence condition. This already implies that the congruence condition Mathematics Subject Classification.
Key words and phrases. surface singularity, rational homology sphere, abelian cover, orbifolds. might not be needed, moreover the following splice diagram ◦ ◦
Γ = ◦ RRRRRRR lllllll
23 15 ◦ lllllll RRRRRRR ◦ ◦ have no manifolds with it as its splice diagram satisfying the congruence condition,even though it satisfy the semigroup condition and it is the splice diagram of dif-ferent manifolds. Nonetheless one can construct a plumbing diagram of the abeliancover use the algorithm derived from my proof of Theorem 6.3 in [Ped10a] which isexplained in more detail in [Ped10b] where this example is explicitly constructed,and one can find a dual resolution graph for the resolution of the splice diagramequation of Γ by hand, and it shows that also in this case is the link of the splicediagram singularity the universal abelian cover. This again indicates that the con-gruence condition is not needed. Even more interesting is the next example. Thefollowing splice diagram ◦ ◦ Γ ′ = ◦ QQQQQQQ mmmmmmm
23 15 ◦ mmmmmmm QQQQQQQ ◦ ◦ .satisfy the semigroup condition, and the universal abelian cover is the link of thesplice diagram singularity. But there are no manifolds with Γ ′ as its splice diagram.So what is the link the universal abelian cover of? To prove Theorem 6.6 of [Ped10a]I had to generalize the notion of splice diagram to a class of -dimensional orbifoldswhich I called graph orbifolds, and Γ ′ is the splice diagram of several graph orbifold.This leads to the purpose of this article, to generalize the congruence conditionto graph orbifolds, which is done in Section 6. Show that, under a small extrahypothesis, the link of at splice diagram equation of Γ( M ) is the universal abeliancover of M if M satisfy the orbifold congruence condition in Section 7. In section8 we show that this is indeed an extension of the results of [NW05a], by givenany two nodes splice diagram Γ satisfying the semigroup condition constructing agraph orbifold M satisfying the orbifold congruence condition, with Γ as its splicediagram. Section 2 introduces graph orbifolds, Section 3 introduces splice diagram,and Section 4 the splice diagram equations. Section 5 introduces the discriminantgroup which is needed in the definition of the congruence condition.2. Graph Orbifolds
To generalize the conditions for the splice diagram equation to define the univer-sal abelian cover, we have to extend the notion of splice diagrams to graph orbifolds,so in this section we define graph orbifolds.
Definition 2.1. A graph orbifold is a 3-dimensional orbifold M , in which there exista finite collection { T i } i ∈ ,...,n of smoothly embedded tori, such that M − S ni =1 T i isa collection of S orbifold fibrations over orbifold surfaces.We will only consider graph orbifolds which has compact closure which is also agraph orbifold, and the boundary components will always be smoothly embeddedtori. Notice that if M is smooth then M is a graph manifold, since any S orbifoldfibration over an orbifold surface with smooth total space is Seifert fibered.Next we want to describe how a graph orbifold look locally, so we will lookat a S orbifold fibration over orbifold surface π : M → Σ . If x ∈ Σ is not anorbifold point then there is a disk neighborhood D of x such that M | π − ( x ) is atrivial fibered solid torus. So the interesting situation is when x ∈ Σ is an orbifoldpoint. This means that a neighborhood U of x is homeomorphic to R / ( Z /α Z ) for a α > , where the group acts as rotation. Then M | π − ( U ) is homeomorphic to PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 3 ( U × S ) / ( Z /α Z ) where [ k ] ∈ Z /α Z acts by [ k ]( z, s ) = ( e πikv/α z, e − πikq/α s ) wherewe consider U = C , q | α and gcd( v, α ) = 1 . Let β ′ satisfy β ′ v ≡ − α , andlet β = β ′ q . We call the pair ( α, β ) the Seifert invariants of the singular fiber over x if we choose ≤ β < α . Notice if gcd( α, β ) = 1 then the action on ( U × S ) is freeand hence π − ( U ) is smooth. We call q = gcd( α, β ) for the orbifold degree of thesingular fiber. We will say that any non singular fiber has orbifold degree , whichoff course follows from the Seifert invariants of a non singular fiber being (1 , .We denote by N ( α,β ) a solid torus neighborhood of the singular fiber with Seifertinvariants ( α, β ) , and by abuse of notation we call N q a solid torus neighborhoodof a singular fiber of orbifold degree q .One can now construct an unique decomposition of a compact graph orbifold M ,the following way. Let q , . . . , q n be the orbifold degrees of all the singular fiberswhich has orbifold degree greater than , and let N q i be solid torus neighborhoodsof the corresponding singular fiber as above. Then M ′ = M − S ni =1 N q i is a graphmanifold with boundary, and hence the JSJ-decomposition of M ′ gives us a collec-tion of Seifert fibered manifolds with boundary M ′ j ’s. We then makes the piecesof the decomposition of M by gluing back the N q i in their original places. This isunique since the JSJ decomposition of M ′ is unique. We will call this decompositionfor the JSJ-decomposition of M .Let K ⊂ M be an orbifold curve with Seifert invariants ( α, β ) , then the gluingof N ( α,β ) into M ′ = M − N ( α,β ) is completely determined by a simple closed curvein T = ∂M ′ of slope β/α . Let q = gcd( α, β ) be the orbifold degree of K and let α ′ = α/q and β ′ = β/q . Then ( α ′ , β ′ ) determines a Seifert fibration of the solidtorus N ( α ′ ,β ′ ) , notice that the Seifert fibration of N ( α ′ ,β ′ ) and the orbifold structureon N ( α,β ) determines the same closed curve in ∂M ′ since β ′ /α ′ = β/α , thereforeif M K = M ′ S N ( α ′ ,β ′ ) using the same gluing, M K and M has the same topology,even though M and M K is not equal as graph orbifolds since the singular fiber K has different Seifert invariants. If one replaces M with M K i for all curves K i whichhas orbifold degree greater than one, we get a graph manifold M which have thesame topology as M , we call M the underlying manifold of M .In the definition of splice diagram of a graph manifold M , one uses the orderfor the first singular homology group of some graph manifolds constructed from M . Now the above implies that if we just use the singular homology groups whenwe define the splice diagram of a graph orbifold M , then one just get the splicediagram of the underlying manifold. So if using orbifold to extend the theoryhas to have any meaning, we need another homology theory, which reduces tosingular homology in M is smooth, but sees the orbifold structure otherwise. Thisis going to be what we will call orbifold homology . Since we only need the firstorbifold homology group, it suffices to define H orb ( M ) to be the abelianazation ofthe orbifold fundamental group π orb ( M ) . The orbifold fundamental group is a moreclassical object, and have been studied a lot, especially in the geometrization of dimensional orbifolds (see e.g. [Sco83]). Since π orb ( M ) = π ( M ) if M is smooth itfollows that H orb ( M ) = H ( M ) if M is smooth. Proposition 2.2. H orb ( T α,β ) = Z ⊕ Z /q Z where q is the orbifold degree of thecentral fiber, i.e. gcd( α, β ) .Proof. Since π orb ( T α,β ) can be presented as h q, h, t | q α h β = 1 , q α ′ h β ′ = t, [ q, h ] =1 i where αβ ′ − βα ′ = q , it is not hard to show that that is indeed a presentation of Z ⊕ Z /q Z . (cid:3) This can then be used to show the following relating orbifold homology of M and M K defined above which is Proposition 5.2 of [Ped10a] PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 4
Proposition 2.3.
Let K ∈ M be an orbifold curve of degree q , then | H orb ( M ) | = q | H orb ( M k ) | . It then follows that | H orb ( M ) | = Q | H orb ( M ) | = Q | H ( M ) | , where Q = Q K q K with the product is taking over all orbifold curves K of orbifold degree q K . Itshould be mentioned that our H orb ( M ) is part of a homology theory for orbifoldsin general see [ALR07]. 3. Splice Diagrams A splice diagram is a tree with no vertices of valence , which is decorated bysigns on vertices who have valence greater than , we call such vertices nodes , andnon negative integer weights on edges adjacent to nodes. We will call vertices ofvalence for leaves , and will in general not distinguish between a leaf and the edgeconnecting the leaf to a node.We will now explain how to assign a splice diagram as an invariant to any graphorbifold. Let M be a rational homology sphere ( Q HS) graph orbifold, we theconstruct the splice diagram Γ( M ) , by first taking a node for each piece of the JSJ-decomposition of M , the connect two nodes if the corresponding S -fibered piecesof the decomposition are glued to create M . We will in general not distinguishbetween a node in Γ( M ) and the corresponding S -fibered piece. This will resultin a tree since the pieces of the JSJ -decomposition of M corresponds to the piecesof the JSJ -decomposition of the underlying manifold M , and since M is a Q HS itfollows from the comments after Proposition 2.3, that M is a Q HS, and hence its
JSJ -decomposition gives a tree like structure. We add a leaf to a node for eachsingular fiber of the node.Next we want to add the decorations. First the signs at a node v is going to bethe sign of the linking number of two non singular fibers at v , for precise definitionof this see section of [Ped10a], there we do only define it for manifolds, but thedefinition carries over to orbifolds, even though one has to be careful since linkingnumber is not going to be symmetric any more in the case of orbifolds. One cancalculate the linking number in the underlying manifold, and it follows from Lemma6.4 that signs will be the same.Last thing to define is the edge weights. Let v be a node and e an edge adjacentto v , then we will do the following construction to define the edge weight d ve adjacent to v on e . Let T ⊂ M be the torus corresponding to the edge e , cut M along T and let M ′ be the piece not containing v . Let M ve = M ′ S T ( S × D ) by gluing a meridian of the solid torus to the image of a fiber of v in ∂M ′ . Then d ve = | H orb ( M ve ) | .The above definition defines a splice diagram where all weight at leaves aregreater than , we call such splice diagram reduced , we will not always assume thatour splice diagram are reduced, i.e. we will sometimes allow weights at leaf to be . A leaf of weight will correspond to a non singular fiber, so if one has a nonreduced splice diagram of a graph orbifold M , one gets the reduced splice diagramof M by removing all leaves of weight .A general edge between two nodes in a splice diagram looks like... v ◦ n MMMMMMMM n k qqqqqqqq r r v ◦ n qqqqqqqq n k MMMMMMMM ... .to such an edge we associate a number called the edge determinant , which we defineas r r − ε ε (cid:0) Q k i =1 n oi (cid:1)(cid:0) Q k j =1 n j (cid:1) , where ε i is the sign on the i ’th node. This PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 5 number is important since it helps determining which graph manifolds arises assingularity links, by the following theorem from [Ped10a]
Theorem 3.1.
Let M be a Q HS graph manifold with splice diagram Γ( M ) . Then M is the link of an isolated complex surface singularity if and only if, there are nonegative signs on nodes and all edge determinants of Γ( M ) are positive. Not all combinatorial splice diagram arises as the splice diagram of a graphorbifold. We will next introduce an important condition which the splice diagramof any graph orbifold satisfy. Let Γ be a splice diagram, and let v and w be twovertices in Γ . Then one defines the linking number l vw of v and w to be the productof all edge weights adjacent to but not on the shortest path from v to w . Similarly l ′ vw is defined the the same way except we exclude the weights adjacent to v and w .Let e be an edge at v then Γ ve is the connected subgraph of Γ one get by removing v , which includes the edge e . We then define ideal generator at v in direction of ed ve , to be the positive generator of the following ideal in Z h l ′ vw | w is a leaf of Γ ve i Definition 3.2.
A splice diagram Γ satisfy the ideal condition if every edge weight d ve is divisible by the corresponding ideal generator d ve In section 12.1 in [NW05a] Neumann and Wahl proves that every splice diagramcoming from a singularity link satisfy the ideal condition, and the proof also worksin the general setting of any graph orbifold. So satisfying the ideal condition is anecessary condition for a splice diagram to come from graph orbifold, in the onenode case the ideal condition is void and we show in section 8 the in the two nodecase the ideal condition is also sufficient, unfortunately this is not the case if thesplice diagram have more than two nodes, and the lack of understanding whichsplice diagram that in general arises as invariant of graph orbifolds is one of thereasons we have not been able to extend the main result to more than two nodessplice diagrams. 4.
Splice Diagram Equations
We are in general going to be be interested in splice diagram satisfying thefollowing stronger condition than the ideal condition.
Definition 4.1.
A splice diagram Γ is said to satisfy the semigroup condition iffor every node v and edge e at v . The edge weight lies in the following semigroupof N : d ve ∈ N h l ′ vw | w a leaf in Γ ve i The semigroup is only interesting if one has no negative signs at nodes, so we willassume that splice diagrams satisfying the semigroup satisfy this. The semigroupcondition is strictly stronger than the ideal condition for splice diagram with morethan one node.If M satisfy the semigroup condition, then given a node v and adjacent edge e ,one can write the corresponding edge weight as d ve = X w is a leaf of Γ ve α vw l ′ vw , (1)where the α vw ’s are non negative integers. We call the collection of α vw semigroupcoefficients of d ve . It not hard to see that (1) is equivalent to d v = X w is a leaf of Γ ve α vw l vw , (2) PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 6 where d v is the product of all edge weights adjacent to v .From now on we will assume that Γ satisfy the semigroup condition, we thenassociate to each leaf w of Γ a variable z w , and have the following definitions. Definition 4.2.
Let v be a node of Γ and assume Γ has leaves w , . . . , w n , then a v - weighting , or v - filtration , of C [ z w , . . . , z w n ] is to associate the weight l vw i to z w i . Definition 4.3.
Let v be a node of Γ an e an adjacent edge, then an admissiblemonomial associated to v and e , is a monomial on the form M ve = Q w z α w w , wherethe product is taken over all leaves in Γ ve , and the α w ’s is a choice of semigroupcoefficients of d ve .It is clear that each admissible monomial is v -weighted homogeneous, of total v -weight d v . Definition 4.4.
Let Γ with leaves w , . . . , w n be a splice diagram satisfying thesemigroup condition, then a set of splice diagram equations for Γ , is the followingset of equations in the variables z w , . . . , z w n . X e a vie M ve + H vi , v a node of valence δ v , e an adjacent edge , i = 1 , . . . , δ v − where • M ve is an admissible monomial • for every v , all maximal minors of the (( δ v − × δ v ) -matrix ( a vie ) has fullrank • H vi is a convergent power series in the z w i ’s all of whose monomials has v -weight higher that d v .This defines n − equations in n variables, and the corresponding subscheme X (Γ) ⊂ C n is called a splice diagram surface singularity .We have the first important result concerning splice diagram equations, which isTheorem 2.6 of [NW05a]. Theorem 4.5.
Let Γ be a splice diagram satisfy the semigroup condition, and let X = X (Γ) be an associated splice diagram surface singularity. Then X is a two-dimensional complete intersection, with an isolated singularity at the origin. Plumbing and The Discriminant Group
From now M will always be a graph orbifold with splice diagram Γ( M ) and M is the underlying manifold, hence Γ( M ) is equal to Γ( M ) whit any edge-weight d ve ,replaced by d ve /o o · · · o n , where o , . . . , o n are the orbifold degrees of all orbifoldcurves in M ve by Proposition 2.3,Now this of course do not always produce an reduced splice diagram of M , sincethere could be leafs with weight . Example 5.1.
Assume M is a graph manifold with following splice diagram ◦ w ◦ ◦ kkkkkkk PPPPPPP Γ( M ) = ◦ PPPPPPP nnnnnnn ◦
102 4755 nnnnnnn PPPPPPP w ◦◦ w ◦ ,and assume that at the leaves named w , w and w , we have orbifold curves oforbifold degrees , and respectively. Then M is going to have the following PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 7 splice diagram ◦ w ◦ ◦ kkkkkkk OOOOOOO Γ( M ) = ◦ OOOOOOO ooooooo
433 26 ◦
56 317 ooooooo OOOOOOO w ◦◦ w ◦ .If we want the reduced splice diagram of M , we just remove the leaf w , and the thecentral vertex becomes a valence two vertex, and therefore has to be suppressed,and we get w ◦ ◦ Γ ′ ( M ) = ◦ OOOOOOO ooooooo
433 317 ◦ ooooooo OOOOOOO ◦ w ◦ .Let ∆( M ) be a plumbing diagram of M , then one can get a plumbing diagram ∆( M ) for M , by adding a arrow weighted with the orbifold degree at the corre-sponding vertex for each orbifold curve of M . By blowing up if necessary, we canassume that each vertex has at most one arrow attached. If v is a vertex of ∆( M ) ,then the orbifold degree o v of v is the orbifold degree of the arrow attached to v , ifno arrow is attached to v then o v = 1 . When ever we use that notation ∆( M ) and ∆( M ) , we will assume that they are connected in this way. Example 5.2.
The graph orbifold M from Example 5.1 has the following plumbingdiagram − ◦ − ◦ llllll RRRRRR − ◦ (3) k k WWWWWWWWW − ◦ llllll − ◦ (2) ' ' OOOOOO ∆( M ) = − ◦ OOOOOOoooooo − ◦ − ◦ − ◦ oooooo OOOOOO − ◦ ggggggggg − ◦ (5) ) ) RRRRRR − ◦ . Let E v ⊂ X be the curve where X is an analytic surface with ∂X = M cor-responding to the vertex v ∈ ∆( M ) and E v the corresponding surface in X with ∂X = M . Then let E : = M v ∈ vert (∆( M )) Z · E v (3)and E : = M v ∈ vert (∆( M )) Z · E v (4)Now E and E are the same as Z -modules, but they have different intersectionpairings defined by ∆( M ) and ∆( M ) . Let A ( M ) be the intersection matrix of E in the basis given by E v . Then the intersection matrix for E in the basis E v isgiven by the matrix A ( M ) which is gotten from A ( M ) by multiplying each column,which corresponds to a vertex v , with the orbifold degree o v of v as defined above.One can construct Γ( M ) from ∆( M ) by suppressing all vertices of valence in ∆( M ) to get the tree structure, and using the following propositions from [Ped10a]to get the decorations. Proposition 5.3.
Let v be a node in Γ( M ) , and e be a edge on that node. We getthe weight d ve on that edge by d ve = | det( − A (∆( M ) ve )) | , where ∆( M ) ve is is theconnected component of ∆( M ) − e which does not contain v . PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 8 ∆( M ) = ... va vv ◦ U U U U U Ui i i i i i e a ww ◦ eeeeeeeee YYYYYYYYY ... | {z } ∆( M ) ve Proposition 5.4.
Let v be a node in Γ( M ) . Then the sign ε at v is ε = − sign( a vv ) ,where a vv is the entry of A ( M ) − corresponding to the node v . Let { e v } ⊂ E ∗ = hom( E , Z ) ⊂ E ⊗ Q and { ˜ e v } ⊂ E ∗ = hom( E , Z ) ⊂ E ⊗ Q bethe dual bases.Define the discriminant group as the finite abelian group D (∆( M )) := E ∗ / E , the order of D (∆( M )) is det( M ) : = | det( − A ( M )) | . The intersection pairing of ∆( M ) induces pairings of E ⊗ Q into Q and D ( M ) into Q / Z . We then get thefollowing facts about discriminant groups from Section 5 of [NW05a]. Proposition 5.5.
Consider a collection e w , where w runs over all leaves of Γ( M ) .Then D ( M ) is generated by the images of these e w . Proposition 5.6.
Let e , . . . , e n be the elements of the dual basis of E ∗ correspond-ing to the leaves of Γ( M ) . Then the homomorphism E ∗ → Q n defined by e ( e · e , . . . , e · e n ) induces an injection D ( M ) ֒ → ( Q / Z ) n . In fact, each non-trivial element of D ( M ) gives an element of ( Q / Z ) n with at leasttwo non-zero entries. We will embed ( Q / Z ) n into C n via the following map ( . . . , r, . . . ) ( . . . , exp(2 πir ) , . . . ) =: [ . . . , r, . . . ] . Proposition 5.7.
Let w , . . . , w n be the leaves of Γ( M ) , then the discriminantgroup D ( M ) is naturally represented by a diagonal action on C n , where the entriesare n -tuples of | det( M ) | ’th roots of unity. Each leaf w j corresponds to an element (cid:2) e w j · e w , . . . , e w j · e w n (cid:3) := (cid:0) exp(2 πie w j · e w ) , . . . , (exp(2 πie w j · e w n ) (cid:1) , and any n − of these generate D ( M ) . The representation contains no pseudore-flections, i.e. non-identity elements fixing a hyperplane. Congruence Condition for Graph Orbifolds
The plumbing diagram ∆ we use will be assumed to be quasi-minimal , this meansthat all weights on strings of ∆ have weights less that − , unless the string consistof a single vertex with weight − .To any string − b ◦ − b ◦ _____ − b k ◦ in ∆ one associates a continued fraction n/p := b − b − b − . . .. We associate / to the empty string. We need the following standard facts aboutthis relation ship which proofs are not hard and can be found many places. PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 9
Lemma 6.1.
Reversing a string with continued fraction n/p gives one with con-tinued fraction n/p ′ with pp ′ ≡
1( mod n ) . Moreover the following relations holds: n = det (cid:0) − b ◦ − b ◦ _____ − b k ◦ (cid:1) p = det (cid:0) − b ◦ − b ◦ _____ − b k ◦ (cid:1) p ′ = det (cid:0) − b ◦ − b ◦ _____ − b k − ◦ (cid:1) ,and the continued fraction in the last case is p ′ /n ′ with n ′ = ( pp ′ − /n .For each n/p ∈ [1 , ∞ ] , there is a unique quasi minimal string, and in this casethe continued fraction associated to the reverse direction is n/p ′ where p ′ is theunique number satisfying p ′ ≤ n and pp ′ ≡
1( mod n ) . As we saw in last section the discriminant group D ( M ) acts diagonally on C n .Viewing the z w i ’s as linear functions on C n , D ( M ) act naturally on C [ z w , . . . , z w n ] ,where e acts on a monomial as Y z α wi w i h − X ( e · e w i ) i Y z α wi w i . This is the same as saying that the group transforms each monomial according tothe character e exp (cid:16) − πi X ( e · e w j ) α w j (cid:17) . Now we will return the setting of Γ( M ) satisfying the semigroup conditions, sowe have the notion of admissible monomial. Definition 6.2.
Let M be a graph orbifold with splice diagram Γ( M ) satisfythe semigroup condition. Let ∆( M ) be a plumbing diagram of M , then ∆( M ) satisfy the orbifold congruence condition if for each node v of Γ( M ) and adjacentedge e , one can choose admissible monomials M ve so that D ( M ) transforms thesemonomials according to the same character.Notice that if M is a manifold, then this definition is the same as the definitionof congruence condition (Definition 6.3) in [NW05a]. Next we write down explicitequation in terms of Γ( M ) and ∆( M ) for the congruence condition. Lemma 6.3.
The matrix ( e v · e v ′ ) where v, v ′ ∈ vert(∆( M )) is the inverse matrixof A ( M ) , and the matrix ( e v · e v ′ ) is the inverse matrix of A ( M ) Proof.
This follows from elementary linear algebra. (cid:3)
Lemma 6.4.
For any v, v ′ ∈ vert(∆( M )) , we have that e v · e v ′ = e v · e v ′ o v ′ . (5) Proof.
This follows from Cramer’s rule, i.e. if M ij is the i, j ’th minor of A ( M ) and M ij is the i, j ’th minor of A ( M ) , then M ij is M ij with each column l multipliedby the corresponding orbifold degree o l , hence det( M ij ) = (cid:0) Q l = j o l (cid:1) det( M ij ) and A ( M ) − i,j = 1det( A ( M )) ( − i + j det( M ij )= Q l = j o l Q l o l A ( M )) ( − i + j det( M ij ) = 1 o j A ( M ) − i,j . (cid:3) Lemma 6.5. If v and v ′ are different vertices in ∆( M ) , corresponding to leaves of Γ( M ) , then e v · e v ′ = − o v l vv ′ d . (6) PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 10
Proof. e v · e v ′ = − ˜ l vv ′ ˜ d by Lemma 6.4 in [NW05a], but by the definition of M itfollows that l vv ′ = l vv ′ Q v ′′ = v,v ′ o v ′′ and d = d Q v ′′ o v ′′ , so using lemma 6.3 we get e v · e v ′ = e v · e v ′ o v ′ = − o v ′ l vv ′ d = − o v ′ l vv ′ Q v ′′ = v,v ′ o v ′′ Q v ′′ o v ′′ d = − o v l vv ′ d . (cid:3) Proposition 6.6. If v is a node in Γ( M ) and w is a adjacent leaf and the continuedfraction associated to the string in ∆( M ) (and ∆( f M ) ) is given by ( n w /o w ) /p , where n w is the weight of the leaf and o w is the orbifold degree of the leaf. Let p ′ be thesmallest positive integer such that p ′ p = 1 mod ( n w /o w ) . Let { n i } ki =1 be the otherweights adjacent to v and let N = Q ki =1 n i . Then e w · e w = − o w Ndn w − p ′ n w . (7) Proof.
By using 6.3 and the formula for e w · e w given by proposition 6.6 in [NW05a]we get e w · e w = e w · e w o w = − o w (cid:16) n w /o w Q ki =1 n i /o i ( n w /o w ) d/o i Q i o i + p ′ n w /o w (cid:17) = − o w n w Q i n i n w d − p ′ n w . (cid:3) Corollary 6.7.
The class of e w ′ , where w ′ is a leaf, transforms the monomial Q z α w w by multiplication by the root of unity h X w = w ′ α w o w ′ l ww ′ det(Γ) − α w ′ e w ′ · e w ′ i . (8)We are now able to give formulas for checking the congruence condition. Proposition 6.8.
Let ∆ be an orbifold plumbing diagram which splice diagram Γ satisfy the semigroup condition. Then the orbifold congruence condition is equiva-lent to the following: for every node v and adjacent edge e , there is an admissiblemonomial M ve = Q z α w w where w is a leaf in Γ ve , so that for every leaf w ′ of Γ ve , h X w = w ′ α w o w ′ l ww ′ det(Γ) − α w ′ e w ′ · e w ′ i = h o w ′ l vw ′ det(Γ) i . (9) Proof.
The proof follows exactly as the prof given for the manifold case in Propo-sition 6.8 in [NW05a]. (cid:3)
We will now look closer to how the congruence condition looks in the two nodecase. Let Γ be the splice diagram w ◦ w ◦ Γ = ... v ◦ n KKKKKKKK n k ssssssss r r v ◦ n ssssssss n k KKKKKKKK ... w k ◦ w k ◦ .and let the orbifold degree corresponding to n ij be o ij . let ∆ be a plumbing diagramwhere we have made sure by blowing up that all the vertices v with o v = 1 havevalence one. Then ∆ is going to look like PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 11 ◦ ( o ) k k WWWWWWW ◦ ( o ) iiiiii ∆ = ... − b ◦ __________ ←−−−−−−− ( n o ) /p W W W W W W W W ←−−−−−−−−− ( n k o k ) /p k g g g g g g g g −−→ n/p − b ◦ −−−−−−−→ ( n o ) /p gggggggg −−−−−−−−−→ ( n k o k ) /p k WWWWWWWW ... ◦ ( o k ) s s ggggggg ◦ ( o k ) ) ) TTTTTT .Let p ′ ij be the unique positive integer satisfying p ij p ′ ij ≡ n ij /o ij ) and p ′ ij < ( n ij /o ij ) . Let N i = Q j n ij . Then the equations for the congruence conditionbecomes h o lj N N n lj d i = h k l X i =1 i = j α w li o lj N l r l n li n lj d − α lj e w lj · e w lj i = h k l X i =1 i = j α w li o lj N l r l n li n lj d − α lj (cid:0) − o lj N l r l n lj n lj − p ′ lj n lj (cid:1)i = h o lj r r n lj d + p ′ lj n lj i Where we use that r − l = P k l i =1 α li N l n li be the choice of admissible monomials. Thisequality is of course equivalent to h i = h o lj ( r r − N N ) n lj d + p ′ lj n lj i = h o lj nn lj + p ′ lj n lj i where we use that nd = r r − N N by the edge determinant equation (Corollary3.3 in [Ped10a]. Since n lj /o lj is an integer, the equation becomes equivalent to thefollowing α lj o lj p ′ lj ≡ − n mod ( n lj /o lj ) . Using the definition of p ′ ij gives us the following set of equations one need to checkfor the congruence to be satisfied α lj o lj ≡ − np lj mod ( n lj /o lj ) . (10)7. Splice diagram equations determining the universal abelian cover
Let M be a graph orbifold with splice diagram Γ = Γ( M ) satisfying the semi-group condition. Assume that { α w } is a choice of semigroup coefficients such that ∆( M ) satisfying the orbifold congruence condition. Definition 7.1.
The set of semigroup coefficients { α w } is said to be M reducible if the orbifold degree o w divides α w for each leaf w ∈ Γ .Let M be the underlying manifold of M , then the splice diagram Γ = Γ( M ) isequal to Γ except that given an edge e ∈ Γ at a node v , then the edge weight d ve in Γ is given by d ve = d ve Q w ∈ Γ ve o w where d ve is the edge weight in Γ . If { α w } is a seton M reducible semigroup coefficients then { α w } , where α w = α w /o w , is a set ofsemigroup coefficients for M since d ve = d ve Q w ∈ Γ ve o w = P w ∈ Γ ve α w l ′ vw Q w ∈ Γ ve o w = X w ∈ Γ ve α w /o w l ′ vw Q w ′ ∈ Γ ve w ′ = w o w ′ = X w ∈ Γ ve α w l ′ vw PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 12
Proposition 7.2.
Assume that { α w } satisfy the orbifold congruence condition for ∆( M ) then { α w } satisfy the congruence condition for ∆( M ) .Proof. Let v ∈ Γ be a node and w ′ ∈ Γ ve be a leaf, then h X w = w ′ α w l ww ′ det( M ) − α w ′ e w ′ · e w ′ i = h X w = w ′ α w /o w l ww ′ Q v ′ = w,w ′ o v ′ Q v ′ o v ′ det( M ) − α w ′ o w ′ o w ′ e w ′ · e w ′ i = h X w = w ′ o w ′ α w l ww ′ det( M ) − α w ′ e w ′ · e w ′ i = h o w ′ l vw ′ det( M ) i = h o w ′ l vw ′ Q v ′ = w ′ o v ′ det( M ) Q v ′ o v ′ i = h l vw ′ det( M ) i . Where we use that { α w } satisfy the orbifold congruence condition to get from thethird line to the fourth. (cid:3) Associate to each leaf w ∈ Γ a variable z w , for each node v ∈ Γ let { α vw } be a reducible choice of semigroup coefficients satisfying the orbifold congruencecondition, and let α vw = α vw o w . Let Z ve = Q w ∈ Γ ve z α vw w and Z ve = Q w ∈ Γ ve z α vw w .Let t be the number of leaves of Γ and Let V be the subvariety of C t defined bythe equations Σ e a vie Z ve = 0 , v a node, i = 1 , . . . , δ v − , where for all v the maximal minors of the (( δ v − × δ v ) -matrix ( a vie ) have maximalrank. Likewise let V be the subvariety of C t defined by the equations Σ e a vie Z ve = 0 , v a node, i = 1 , . . . , δ v − , with the same choice of a vie as for V . Define the map F : C t → C t by F ( z w , . . . , z w t ) = ( z o w w , . . . , z o wt w t ) . (11)Then F ( V ) = V , and F is a branched abelian cover of C t with deg( F ) = Q w o w ,branched over B = S w { z w = 0 | o w > } . Now F | V : V → V is a branched abeliancover, branched over V T B .Let X be a singularity which has resolution ∆( M ) and hence has link M . Sincethe equations for V satisfy the congruence conditions for ∆( M ) , V defines theuniversal abelian cover of X branched over the origin by the work of Neumann andWahl [NW05a]. Let π : V → X denote the covering map, then deg( π ) = | H ( M ) | = | H orb ( M ) | Q w o w .Now M embeds into X , so choose a small enough embedding i and let L ( V ) = π − ( i ( M )) , then L ( V ) is homeomorphic to the link of V . Let L ( V ) = F − ( L ( V )) ,then by choosing small enough embedding L ( V ) is homeomorphic to the link of V .Then the restrictions of the maps F | L ( v ) : L ( V ) → L ( V ) and i − ◦ π | L ( V ) : L ( V ) → M are abelian covers, the first branched over L ( V ) T B .Let f : M → M be the homeomorphism which identifies M and M as topologicalspaces. Definition 7.3.
Let π : L ( V ) → M be defined as π = f ◦ i − ◦ π | L ( V ) ◦ F | L ( V ) .A priori π is just a continues map, we next turn to prove that π is an abelianorbifold cover.Let K w ⊂ M be the singular fiber corresponding to the leaf w ∈ Γ . Let S bethe singular set of M , then S = S w : o w > K w . We want to determine π − ( S ) . First PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 13 f − ( S ) = S where we see S as a subspace of M , i.e. S is the union of singularfibers of M for which o w > , the last information is not detectable from M . Now π | − L ( V ) ( K w ) = L ( V ) T { z w = 0 } and hence ( f ◦ i − ◦ π | L ( V ) ) − ( S ) = L ( V ) \ [ w : o w > { z w = 0 } = L ( V ) \ B Now f : ( M − f − ( S )) → ( M − S ) is an abelian cover, since f is the identity awayfrom S , π | L ( M ) : ( L ( M ) − B ) → ( M − f − ( S )) is an abelian cover, since it is therestriction of an abelian cover to a union of fibers. And F | L ( V ) : ( L ( V ) − F − ( B )) → ( L ( M ) − B ) is an abelian cover, since it is the restriction of a branched abeliancover to the compliment of the branched locus. Hence π | ( L ( V ) − π − ( S )) : ( L ( V ) − π − ( S )) → ( M − S ) is an abelian cover of degree deg( F ) deg( π ) deg( f ) = ( Q w o w ) | H orb ( M ) | Q w o w | H orb ( M ) | .So next we turn to what happens in the neighborhood of an orbifold curve. Proposition 7.4.
Let K w ⊂ M be a orbifold curve of degree o w and let N K w be asolid torus neighborhood of K w , then π | π − ( N Kw ) : π − ( N K w ) → N K w is an abelianorbifold cover of degree | H orb ( M ) | .Proof. We both need to show that the exist open set U ⊂ R and D ⊂ R and abranched abelian cover ˜ π : U → D × S and a homomorphism ψ : Z /a Z → Z /α Z such that π is equivariant with respect to ψ and the following diagram commutes U ˜ π / / (cid:15) (cid:15) D × S (cid:15) (cid:15) U/ ( Z /a Z ) (cid:15) (cid:15) ( D × S ) / ( Z /α Z ) (cid:15) (cid:15) π − ( N K w ) π / / N K w . The vertical maps are the ones given from the orbifold structures of π − ( N K w ) and N K w , i.e. the upper maps are quotient maps and lower maps homeomorphisms. Wecan choose D to be a disk, and by choosing it small enough (i.e. choosing N K w smallenough) we get that π − ( N K w ) is a disjoint union of solid torus neighborhoods V K of an singular fiber K , hence we can choose U to be a disjoint unions of D × S .We now only need to see that the diagrams commute for each of these component.Let t n = e πi/n , then Z /α Z acts on ( D × S ) by ( x, s ) → ( t v α α x, t o w α s ) where gcd( α, v α ) = 1 . Likewise Z /a Z acts on ( D × S ) by ( x, s ) → ( t v a a x, t a s ) .Let N ′ k = f − ( N K w ) ⊂ M , and then π − ( N ′ K ) = S mi =1 V ′ K , where V ′ K is asolid torus neighborhood of a singular fiber of degree a ′ . Let α ′ = α/o w and β ′ = β/o w then the orbifold structure on N ′ K is given by Z /α ′ Z acting on D × S by ( t v α ′ α ′ x, t α ′ s ) , where v α ′ β ′ ≡ − α ′ and this implies that v α ≡ v α ′ mod α ′ .Since π is an abelian orbifold cover there exist an homomorphism ψ ′ : Z /a ′ Z → Z /α ′ Z , and an abelian cover π ′ such that the following diagram commutes D × S π ′ / / (cid:15) (cid:15) D × S (cid:15) (cid:15) ( D × S ) / ( Z /a ′ Z ) (cid:15) (cid:15) ( D × S ) / ( Z /α ′ Z ) (cid:15) (cid:15) V ′ K π / / N ′ K , PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 14 and ψ ( t a ′ ) π ′ ( x, s ) = π ′ ( t v a ′ a ′ x, t a ′ s ) .Now since π is an abelian cover, it follows that α ′ = λa ′ for some λ and hence, ψ ′ ( t ′ a ) = t λα ′ .Now restricting F to V K one sees that F is a branched cover and if V K = D × S then F ( x, s ) = ( x o w , s ) ∈ V ′ K = D × S . Now the Seifert fibered structure on V ′ K is define by the curve of slope b ′ /a ′ , and it lifts to the curve of slope o w b ′ /a ′ andhence a = a ′ and b = o w b ′ and hence v a ′ ≡ o w v a mod a ′ .We can now define ˜ π and ψ . ˜ π ( x, s ) = π ′ ( x o w , s ) and ψ ( t a ) = t o w λα . So first weneed to check that ˜ π is equivariant with respect to ψψ ( t a ) e π ( x, s ) = t o w λα π ′ ( x o w , s ) = ψ ′ ( t ′ a ) π ′ ( x o w , s ) = π ′ ( t v a ′ a ′ x o w , t a ′ s )= π ′ ( t o w v a a ′ x o w , t a ′ s ) = π ′ (( t v a a x ) o w , t a ′ s ) = ˜ π ( t v a a x, t a s ) So what is left is just checking that the diagram commutes. Start by taking ( x, s ) ∈ D × S , then the one composition is taking ˜ π ( x, s ) and send it to the classin ( D × S ) / ( Z /α Z ) , and the other composition is sending ( x, s ) into the class in ( D × S ) / ( Z /a Z ) and then take π . If we denote the class in ( D × S ) / ( Z /k Z ) where the action is given by the integers k, l by [ x, s ] ( k,l ) , then we need to see that π ([ x, s ] ( a,b ) ) = [˜ π ( x, s )] ( α,β ) . Now π ([ x, s ] ( a,b ) ) = f ( π ( F ([ x, s ] ( a,b ) ))) by definition.Since the Seifert fibered structure on V K is given by pulling back the Seifert fiberedstructure on V ′ K by ( x o w , s ) , F ([ x, s ] ( a,b ) ) = [ x o w , s ] ( a ′ ,b ′ ) . By construction we havethat π ([ x, s ] ( a ′ ,b ′ ) ) = [ π ′ ( x, s )] ( α ′ ,β ′ ) , and by definition f ([ x, s ] ( α ′ ,β ′ ) ) = [ x, s ] ( α,β ) ,hence π ([ x, s ] ( a,b ) ) = [ π ′′ ( x o w , s )] ( α,β ) , and the diagram commutes.Last we need to calculate the degree of π | U . First the degree of π | V K : V K → N K w is o w λ . The degree of π on π − ( N ′ K ) = S mi =1 V ′ K is | H ( M ) | , hence m = | H ( M ) | /λ . U = S ( Q w ′6 = w o w ′ ) i =1 π − ( V ′ k ) and hence the number of component of U is ( | H ( M ) | /λ ) Q w ′ = w o w ′ and deg( π | U ) = o w λ ( | H ( M ) | /λ ) Q w ′ = w o w ′ = | H ( M ) | Q w ′ o w ′ = | H orb ( M ) | . (cid:3) Combining the above results gives us the following theorem
Theorem 7.5.
Let M be a rational homology sphere graph orbifold with splicediagram Γ satisfying the semigroup condition. Suppose there exist a graph orbifold M ′ also with splice diagram Γ , and a set of reducible semigroup coefficients { α } for M ′ satisfying the orbifold congruence condition. Then the link of the completeintersection defined by (Γ , { α } ) is homeomorphic to the universal abelian cover of M .Proof. The above show that π : L ( V ) → M ′ is an orbifold abelian cover of degree | H orb ( M ) | , and hence the universal abelian cover of, M ′ , combining this with thesecond main theorem of [Ped10a] gives the result. (cid:3) So to prove that the the splice diagram always define the universal abelian cover,one just have to show that given M with Γ( M ) satisfying the semigroup condition,there always exist a M ′ with a reducible set of admissible monomials satisfying theorbifold congruence condition such that Γ( M ′ ) = Γ( M ) . We will in the next sectionshow this is always true in the case of a splice diagram with only two nodes, byconstructing such a M ′ from any two node splice diagram satisfying the semigroupcondition.8. Algorithm for construction an orbifold with a given two nodesplice diagram
In this section we will make an algorithm which given any two node splice di-agram Γ satisfying the ideal generator condition gives a graph orbifold M with PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 15 Γ( M ) = Γ . We will construct M by giving a plumbing diagram ∆ such that ∆ = ∆( M ) . We will not give a complete plumbing diagram, but specify the orb-ifold degrees, the weight at the nodes, and the continued fraction associated tothe strings. From this data one can the obtain the complete plumbing diagram ifneeded. Let the splice diagram look like the following ◦ ◦ Γ = ... ε ◦ n KKKKKKKK n k ssssssss r r e ε ◦ n ssssssss n k KKKKKKKK ... ◦ ◦ .Let N j = Q i n ji , and let D be the edge determinant of the central edge. Theplumbing diagram will be given by ◦ ( o ) k k WWWWWWW ◦ ( o ) iiiiii ∆ = ... − b ◦ ←−−−−−−− ( n o ) /p W W W W W W W W ←−−−−−−−−− ( n k o k ) /p k g g g g g g g g − b ◦ −−−−−−−→ ( n o ) /p gggggggg −−−−−−−−−→ ( n k o k ) /p k WWWWWWWW ... ◦ ( o k ) s s ggggggg ◦ ( o k ) ) ) TTTTTT .So we need to specify o ji , p ji and b j , from the information given by Γ . • First chose integer α ji such that r − j = P i α ji N j n ji , these integer existsince Γ satisfy the ideal generator condition. If Γ furthermore satisfy thesemigroup condition, then the α ji ’s can be chosen to be non negative, andthe choice of α ji ’s is a choice of semigroup coefficients for r − j . • Let λ ji be the smallest integer, such that λ ji n ji ≥ ε j α ji if D > , and λ ji n ji ≥ − ε j α ji if D < • Let o ji = gcd( n ji , α ji ) . • Let p ji = λ ji n ji − α ji o ji if D > , and p ji = λ ji n ji + α ji o ji if D < . • Let b j = P i λ ji .Notice that gcd( n ji /o ji , p ji ) = 1 so these choices gives a well-defined plumbing. Proposition 8.1.
Let M be the graph orbifold given by ∆ with the above choices,then Γ( M ) = Γ and | H orb ( M ) | = D ( e ) .Proof. Since the weight to the leaves in Γ( M ) is ( n ji /o ji ) o ji , it is the same weightas in Γ . Next we start by considering the case that D > , then the unnormalizededge determinant equation (Lemma 3.2 in [Ped10a]) implies that det( M ) > , sothe only thing to check is that ˜ r j , the weights associated to the central string in Γ( M ) , is ε − j r j . ˜ r − j = (cid:0) Y i o ji (cid:1) det(∆( f M ) v j +1 e ) = (cid:0) Y i o ji (cid:1)(cid:0) Y i n ji /o ji (cid:1)(cid:0) b j − X i p ji n ji o j i (cid:1) = N j (cid:0) X i λ ji − X i p ji o ji /n ji (cid:1) = X i (cid:0) λ ji N j − λ ji n ji − ε j α ji o ji o ji N j n ji (cid:1) = X i ε j α ji N j n ji = ε j r − j . The case whit
D < is similar, but now det( M ) < and ˜ r j = − ε − j r j . The laststatement follows from the edge determinant equation (Corollary 3.3 in [Ped10a]),since the fiber intersection number of e is , or by using the above calculation tocalculate det(∆) . (cid:3) Notice that M do depend on the choice of α ji ’s. PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 16
Corollary 8.2.
Let Γ be a splice diagram satisfying the semigroup condition. Thenif M is a graph orbifold given by the above algorithm, then M satisfy the orbifoldcongruence condition.Proof. We need to check that the equations α lj o lj ≡ − np ji mod ( n ji /o ji ) given by(10) are satisfied. Now n = 1 so by definition p ji = λ ji n ji − α ji o ji = λ ji n ji o ji − α ji o ji . Which implies that − α lj o lj ≡ p ji mod ( n ji /o ji ) , and hence the congruence conditionis satisfied. (cid:3) Notice that the α ji ’s are a reducible set of semigroup coefficients by the definitionof the o ji ’s. So combining this with Theorem 7.5 we get that given a two node splicediagram Γ satisfying the semigroup condition, the link of any splice diagram surfacesingularity is homeomorphic to the universal abelian cover of any graph orbifoldwith Γ as its splice diagram.Now this method for proving that the link of the splice diagram equations are theuniversal abelian covers does not easily generalize to more the two nodes. Alreadyin the -node case is the semigroup (or ideal) condition not sufficient for a splicediagram to be realized by a graph orbifold. The following diagram ◦◦ ◦ Γ = ◦ RRRRRRR lllllll
11 10 e ◦ e ◦ lllllll RRRRRRR ◦ w ◦ ,is not the splice diagram of any graph orbifold, even though it satisfy the semigroupcondition. The reason is that one has by the edge determinant equation (Corollary3.3 in [Ped10a]) that the order of H orb ( M ) divides all edge determinants, so if Γ where the splice diagram for some M , then | H orb ( M ) | would divide D ( e ) = 26 and D ( e ) = 20 , and hence divide . Now M can not be an integer homology spherebecause then all weight adjacent to a node would have to be pairwise coprimeaccording to [EN85], so H orb ( M ) = Z / Z . Using the topological description of theideal generator given in section 12.1 in [NW05a], one easily sees that given any edge e in Γ( M ) , the product of the two ideal generators associated to each of the ends of e has to divide the order of H orb ( M ) , this includes edges to leaves. Now the idealgenerator associated to leaf w is , and hence do not divide the order of H orb ( M ) so we get a contradiction.The above consideration on ideal generators leads to the following necessarycondition for a splice diagram Γ to be realized from a graph orbifold: The product ofthe ideal generators associated to any edge has to divide all the edge determinants.But even this condition is not sufficient. The following splice diagram satisfy it,but is not realizable by any graph orbifold. ◦ ◦ v ◦ SSSSSSS kkkkkkk Γ ′ = ◦ ◦ ◦ v ◦ SSSSSSS
90 42 kkkkkkkkkkkkkk kkkkkkk v ◦ kkkkkkk SSSSSSSSSSSSSS SSSSSSS ◦ ◦ . PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 17
The edge determinant equation and the above mentioned condition implies that | H orb ( M ) | = 30 if M is a graph orbifold realizing Γ ′ . Then using this and the edgedeterminant equation we can make a splice diagram for M v e since we know that | H orb ( M v e ) | = 90 . We can continue doing this until we get that there exists agraph manifold M ′ with | H ( M ′ ) | = 60 and with the following splice diagram. ◦◦ ssssssss KKKKKKKK ◦ ◦ ,we now this has to be manifold, since all the singular fibers has come from theprocess of creating M ve ’s, and hence does not have orbifold curves. This means M ′ is a Seifert fibered manifold with Seifert invariants (1 , − b ) , (6 , β )(21 , β ) , (10 , β ) ,a simple calculation shows that such a Seifert fibered manifold can not have firsthomology group of order .The failure of the last example is not as easy as the first to specify in a nicecondition, so do at the moment not have a good idea on a set of necessary conditionsfor a splice diagram to be realized by graph orbifold. Even without this, it mightstill be possible to used Theorem 7.5 to prove it for more general graph orbifoldsthat just the once having two node splice diagram.An other interesting question is, what are splice diagram singularities comingfrom diagrams as above. PLICE SINGULARITIES AND UNIVERSAL ABELIAN COVER OF GRAPH ORBIFOLDS 18
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Matematische Institut, Universität Heidelberg, Heidelberg, 69120
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