UNIVERSITÀ DEGLI STUDI DI FIRENZE. Rational homology cobordisms of plumbed manifolds and arborescent link concordance

Transcription

1 UNIVERSITÀ DEGLI STUDI DI FIRENZE Dipartimento di Matematica Ulisse Dini Dottorato di Ricerca in Matematica Rational homology cobordisms of plumbed manifolds and arborescent link concordance Paolo Aceto Tutor Prof. Graziano Gentili Coordinatore del Dottorato Prof. Alberto Gandolfi Relatore Prof. Paolo Lisca CICLO XXVII, SETTORE SCIENTIFICO DISCIPLINARE MAT/03

2 Introduction The study of concordance properties of classical knots and links in the 3-sphere is a highly active field of research in low dimensional topology. Problems in this area involve a wide range of techniques, from the use of sophisticated combinatorial invariants derived from knot homology theories to the interplay with 3 and 4-manifold topology. One of the most famous unsolved problems in this field is the so called slice-ribbon conjecture. A knot K S 3 is smoothly slice if it bounds a properly embedded smooth disk in the 4-ball. A smoothly slice knot is ribbon if the spanning disk D 2 D 4 can be choosen so that there are no local maxima of the radial function ρ : D 4 [0, 1] restricted to the image of D 2. The slice ribbon cojecture states that every slice knot is ribbon. Since it was first formulated by Fox in 1962 (as a question rather than a conjecture) there have been many efforts towards understanding slice and ribbon knots. One stimulating aspect of this topic is that it naturally leads to several related questions on 3-manifold topology. In [13] Lisca proved that the slice ribbon conjecture holds true for 2-bridge knots. He used an obstruction based on Donaldson s diagonalization theorem to determine which lens spaces bound rational homology balls. This technique has been used by Lecuona in [11] to prove that the slice ribbon conjecture holds true for an infinite family of Montesinos knots. In [4] Donald refined the obstruction used by Lisca to determine which connected sums of lens spaces embed smoothly in S 4. The starting point of this work is an adaption of these ideas to the study of slice links with more than one component. The basic idea of [13] can be described as follows. If a knot K is slice its branched double cover Σ(K) is a rational homology sphere that bounds a rational homology ball W. If K is a 2-bridge knot then Σ(K) is a lens space, say L(p, q). Each lens space is the boundary of a canonical plumbed 4-manifold X(p, q) with negative definite intersection form. By taking the union X = X(p, q) W we obtain a smooth closed oriented 4-manifold with unimodular, negative definite intersection form, and by Donaldson s diagonalization theorem this intersection form is diagonalizable over the integers. The inclusion X(p, q) X induces an embedding of intersection lattices (H 2 (X(p, q); Z), Q X(p,q) ) (Z N, I N ). This fact turns out to be a powerful obstruction which eventually leads to a complete list of lens spaces that bound rational homology balls. A link L S 3 is (smoothly) slice if it bounds a disjoint union of properly embedded 1

3 2 disks in the 4-ball, one for each component of L. Let L be a slice link with n components (n>1). The first observation is that Σ(L) is a 3-manifold with b 1 = n 1 which bounds a smooth 4 manifold W with the rational homology of a boundary connected sum of n 1 copies of S 1 D 3 (see Proposition 2.1). Motivated by this fact and focusing on the case n = 2 we are led to the following general problem: Question 0.1. Which rational homology S 1 S 2 s bound rational homology S 1 D 3 s? In Section 2.2 we introduce a general procedure which allows one to construct rational homology cobordisms between plumbed 3-manifolds. For any plumbed 3- manifold Y our procedure gives infinitely many plumbed 3-manifolds which are rational homology cobordant to Y. We then introduce a family F of plumbed 3-manifolds with b 1 = 1. This family includes, up to orientation reversal, all Seifert fibered spaces over the 2-sphere with vanishing Euler invariant. We prove that if a given Y F bounds a rational homology S 1 D 3 then Y can be constructed with our procedure (see Theorem 2.16). This gives us a complete list of the 3-manifolds in F that bound a rational S 1 D 3. By specializing Theorem 2.16 to star-shaped plumbing graphs, we obtain the following characterization for the Seifert fibered spaces over the 2-sphere which bound rational homology S 1 D 3 s. Theorem 0.2. A Seifert fibered manifold Y = (0; b; (α 1, β 1 ),..., (α h, β h )) bounds a QH S 1 D 3 if and only if the Seifert invariants occur in complementary pairs and e(y) = 0. Two pairs of Seifert invariants (α i, β i ) and (α j, β j ) are complementary if they can be chosen so that βi α i + β j α j = 1 (see Section for precise definitions). This result (as well as Theorem 2.16) is obtained by using an obstruction based on Donaldson s theorem. Roughly speaking we proceed as follows. Each Y in F bounds a negative semidefinite plumbed 4-manifold X. If Y bounds a rational homology S 1 D 3, say W, we can form the closed 4-manifold X = X W. The intersection form Q X will be negative definite and this fact provides the constraints we need for our analysis. To each Y F we can associate the family L(Y) of arborescent links whose branched double cover is Y. In general, the family L(Y) contains many non isotopic links. However, these links are all related to each other by Conway mutation. In Section 2.5, as a consequence of Theorem 2.16, we prove the following. Theorem 0.3. Let L be a link in L(Y) for some Y F (e.g. a Montesinos link). The following conditions are equivalent: Y bounds a rational homology S 1 D 3 ; there exists L L(Y) that bounds a properly embedded smooth surface S in D 4 with χ(s) = 2 without local maxima. In particular every 2-component slice link L L(F) has a ribbon mutant.

4 3 The thesis is organized as follows. In Chapter 1 we introduce plumbed manifolds and arborescent links. Most of the results stated there are well known but we need to carefully prepare the language for our work. In Chapter 2 we introduce our construction and state our main results. Chapter 3 contains the combinatorial work which is needed to conclude the proof of Theorem Pictures in Section 2.7 are obtained via the KLO software for Kirby calculus. Acknowledgements I would like to thank my supervisor Paolo Lisca for his support and for suggesting this topic, Giulia Cervia for her constant encouragement and her help in drawing pictures. I would also like to thank Francesco Lin for many stimulating conversations and for suggesting the problem that led to the material contained is Section 2.7. The referees Ana Lecuona and Brendan Owens provided many helpful comments and pointed out several mistakes. I am very grateful to them for their meticulous work which has been of great value for me.

5 Contents 1 Preliminaries Plumbed manifolds The normal form of a plumbing graph The continued fraction of a plumbing graph Reversing the orientation Lens spaces and Seifert manifolds The linear complexity of a tree Arborescent links Montesinos links Homology cobordisms and slice links Slice links and branched covers Building plumbed 3-manifolds with prescribed QH-cobordism class Elementary building blocks Main theorem more plumbing graphs The case b 1 > Slice arborescent links A group structure Some homology spheres which bound homology balls Background and motivation Examples Linear subsets Basic definitions and known results Main results and strategy of the proof Irreducible subsets First case: b(s) = Second case: b(s) = Third case: b(s) = Conclusion Orthogonal subsets Conclusion of the proof

6 CONTENTS 5 Bibliography 84

7 List of Figures 1.1 A surgery description for the Seifert fibered manifold (0; b; (α 1, β 1 ),..., (α k, β k )) The plumbing operation An example of an embedded plumbing graph and a diagram of its associated arborescent link. Note that the weights are assigned to the angular sectors rather than to the vertices. 0-weights are omitted Exchanging two edges on an embedded plumbing graph via Conway mutation On the right a movie description for the properly embedded smooth surface S S 3 I. On the left each diagram is a surgery description for the 3-manifold obtained as the branched double cover along the corresponding link on the right The thick curve on the leftmost diagram is homologous to the sum of the two thick curves on the rightmost diagram The knots 5 2 and 7 4 from the Rolfsen table Surgery descriptions for PΓ and PΓ h A sequence of blowdowns and isotopies needed to identify (PΓ h) with S(5 2, 1 ) An alteration of the surgery descriptions for PΓ h that results in the 3-manifold S(7 4, 1 ) A 2-handle attachment on PΓ which gives the homology sphere S(P(5, 1, 3), 1). 4 On the left a 1-blowdown has been performed A 2-handle attachment on PΓ that gives the homology sphere S(P(1, 1, 4), 1 )

8 Chapter 1 Preliminaries In this chapter we introduce plumbed manifolds and arborescent links. Most of the results stated here are well known. Our main purpose is to fix notation and several conventions as well as introduce some new terminology that will be useful later on. 1.1 Plumbed manifolds In this section, following [16], [17] and [18], we review the basic definitions and properties of plumbed 3-manifolds. We recall Neumann s normal form of a plumbing graph, and the generalized continued fraction associated to a plumbing graph. We show how these data behave with respect to orientation reversal. We briefly recall the definitions of lens spaces and Seifert manifolds viewed as special plumbed manifolds. Definition 1.1. A plumbing graph Γ is a finite tree where every vertex has an integral weight assigned to it. To every plumbing graph Γ we can associate a smooth oriented 4-manifold PΓ with boundary PΓ in the following way. For each vertex take a disc bundle over the 2-sphere with Euler number prescribed by the weight of the vertex. Whenever two vertices are connected by an edge we identify the trivial bundles over two small discs (one in each sphere) by exchanging the role of the fiber and the base coordinates. We call PΓ (resp. PΓ) a plumbed 4-manifold (resp. plumbed 3-manifold). This definition can be extended to reducible 3-manifolds; if the graph is a finite forest (i.e., a disjoint union of trees) we take the boundary connected sum of the plumbed 4- manifolds associated to each connected component of Γ. Unless otherwise stated, by a plumbing graph we will always mean a connected one, as in Definition 1.1. Every plumbed 4-manifold has a nice surgery description which can be obtained directly from the plumbing graph. To every vertex we associate an unknotted circle framed according to the weight of the vertex. Whenever two vertices are connected by an edge the corresponding circles are linked in the simplest possible way, i.e. like the Hopf link. The framed link obtained in this way also gives an integral surgery 7

9 1.1 Plumbed manifolds 8 presentation for the corresponding plumbed 3-manifold. The group H 2 (P(Γ); Z) is a free abelian group generated by the zero sections of the sphere bundles (i.e. by vertices of the graph). Moreover, with respect to this basis, the intersection form of P(Γ), which we indicate by Q Γ, is described by the matrix M Γ whose entries (a ij ) are defined as follows: a i,i equals the Euler number of the corresponding disc bundle a i,j = 1 if the corresponding vertices are connected a i,j = 0 otherwise. Finally note that M Γ is also a presentation matrix for the group H 1 ( PΓ; Z) The normal form of a plumbing graph We will be mainly interested in plumbed 3-manifolds. There are some elementary operations on the plumbing graph which alter the 4-manifold but not its boundary. Following [16] we will state a theorem which establishes the existence of a unique normal form for the graph of a plumbed 3-manifold. In [16] these results are stated in a more general context. Here we extrapolate only what we need in order to deal with plumbed manifolds. First consider the blow-down operation. It can be performed in any of three situations depicted below. 1. We can add or remove an isolated vertex with weight ε {±1} from any plumbing graph. ε Γ Γ 2. A vertex with weight ε {±1} linked to a single vertex of a plumbing graph can be removed as shown below. From now on we use three edges coming out of a vertex to indicate that any number of edges may be linked to that vertex. a ε a ε 3. Finally, if a ±1-weighted vertex is linked to exactly two vertices it can be removed

10 1.1 Plumbed manifolds 9 as shown below. a ε b a ε b ε Next we have the 0-chain absorption move. A 0-weighted vertex linked to two vertices can be removed and the plumbing graph changes as shown. a 0 b a + b The splitting move can be applied in the following situation. Given a plumbing graph with a 0-weighted vertex which is linked to a single vertex v, we may remove both vertices (and all the corresponding edges) obtaining a disjoint union of plumbing trees. We may depict this move as follows Γ 1 0 a Γ 1 Γ k Γ k Proposition 1.2. [16] Applying any of the above operations and their inverses to a plumbing graph does not change the oriented diffeomorphism type of the corresponding plumbed 3-manifold. Before discussing the normal form of a plumbing graph we need some terminology. A linear chain of a plumbing graph is a portion of the graph consisting of some vertices v 1,..., v k (k 1) such that: each v i with 1 < i < k is linked only to v i 1 and v i+1 v 1 and v k are linked to at most two vertices. A linear chain is maximal if it is not contained in any larger linear chain. A vertex of a plumbing graph is said to be:

11 1.1 Plumbed manifolds isolated if it is not linked to any other vertex 2. final if it is linked exactly to one vertex 3. internal otherwise. Note that isolated and final vertices always belong to some linear chain, while an internal vertex belongs to some linear chain if and only if it is linked to exactly two vertices. Definition 1.3. A plumbing graph Γ is said to be in normal form if one of the following holds 1. Γ = 0 or Γ = 2. every vertex of a linear chain has weight less than or equal to -2. Theorem 1.4. [16] Every plumbing graph can be reduced to a unique normal form via a sequence of blow-downs, 0-chain absorptions, splittings and their inverses. Moreover two oriented plumbed 3-manifolds are diffeomorphic (preserving the orientation) if and only if their plumbing graphs have the same normal form. Remark 1.5. We point out that using this theorem one can specify a certain class of plumbed 3-manifolds simply by describing the shape of the plumbing graph in its normal form. In particular we will see at the end of this section that lens spaces and some Seifert manifolds admit such a description The continued fraction of a plumbing graph In this section, following [17] we introduce some additional data associated to a plumbing graph. As we have seen to any plumbing graph Γ we can associate an integral symmetric bilinear form Q Γ. All the usual invariants of Q Γ will be denoted referring only to the graph. In particular rank, signature and determinant will be denoted respectively by rkγ, (b + Γ, b Γ, b 0 Γ) and detγ. Let (Γ, v) be a connected rooted plumbing graph, i.e. a plumbing graph together with the choice of a particular vertex. If we remove from Γ the vertex v and all the corresponding edges we obtain a plumbing graph Γ v which is the disjoint union of some trees Γ 1,..., Γ k (k is the valency of v). Every such tree has a distinguished vertex v j which is the one adjacent to v. Definition 1.6. With the notation above we define the continued fraction of Γ as cf (Γ) := detγ detγ v Q { }

12 1.1 Plumbed manifolds 11 We put α/0 = for each α Q. Remark 1.7. Note that cf (Γ) depends on the rooted plumbing graph (Γ, v). By abusing notation we do not indicate this dependence explicitely. In the sequel, it will always be clear from the context which vertex has been chosen. Proposition 1.8. [17] If the weight of the distinguished vertex is b Z then k detγ = b detγ v detγ vi detγ j i=1 j i and cf (Γ) = b k i=1 1 cf (Γ i ) Reversing the orientation Let Γ be a plumbing graph in normal form. In this section, following [16], we explain how to compute the normal form for the plumbed manifold PΓ, i.e. PΓ with reversed orientation. We call this plumbing graph the dual graph of Γ and we denote it with Γ. For a vertex v of a plumbing graph which is not on a linear chain we define the quantity c(v) to be the number of linear chains adjacent to v, i.e. the number of vertices belonging to a linear chain that are linked to v. For instance in the graph both the trivalent vertices have c = 2. We indicate with (..., [a],... ) a portion of a string with a -chain of length a > 0, i.e. a linear chain consting of a vertices each with weight. Theorem 1.9. [16] Let Γ be a plumbing graph in normal form. Its dual graph Γ can be obtained as follows. The weight w(v) of every vertex which is not on a linear chain is replaced with w(v) c(v), and every maximal linear chain of the form a 1 a 2 a n

13 1.1 Plumbed manifolds 12 is replaced with b 1 b 2 b m where the weights are determined as follows. If (a 1,..., a n ) = ( [n 0], m 1 3, [n 1], m 2 3,..., m s 3, [ns] ) with n i 0, m i 0 and s > 0. Then (b 1,..., b m ) = ( n 0 2, [m 1], n 1 3,..., n s 1 3, [ms], n s 2). If (a 1,..., a n0 ) = ( [n 0] ) then (b 1 ) = ( n 0 1). The reason why we are interested in this construction of the dual graph of a plumbing graph in normal form will be clear in Chapter 2. Essentially we are trying to detect nullcobordant 3-manifolds using obstructions based on Donaldson s diagonalization theorem. Since the property we want to detect does not depend on the orientation of a given 3-manifold it is natural to examine both a plumbing graph Γ and its dual Γ. Moreover, the normal form is specifically defined to give a plumbing graph that minimizes the quantity b + (Γ) among all plumbing graphs representing PΓ (see [17] theorem 1.2). We now introduce a quantity that will play an important role in the analysis developed in Chapter 3. Definition Let Γ be a plumbing graph in normal form, and let v 1,..., v n be its vertices. We define I(Γ) := n 3 w(v i ). i=1 The following Proposition is proved in [13]. It can also be proved directly using Theorem 1.9. Proposition Let Γ be a linear plumbing graph in normal form. We have I(Γ) + I(Γ ) = Lens spaces and Seifert manifolds We briefly recall the plumbing description for lens spaces and Seifert manifolds.

14 1.1 Plumbed manifolds 13 In this context it is convenient to define a lens space as a closed 3-manifold whose Heegaard genus is 1. The difference with the usual definition is that we are including S 3 and S 1 S 2. It is well known that every lens space has a plumbing graph which is either empty (S 3 ) or a linear plumbing graph and that every linear plumbing graph represents a lens space. It follows from Theorem 1.4 that the normal form of a plumbing graph representing a lens space other than S 3 or S 1 S 2 is a linear plumbing graph a 1 a 2 a n... where a i for each i. It is easy to check that given a linear plumbing graph as above we have 1 cf (Γ) = a 1 =: [a a 2 1 1,..., a n ]. a 3... This fact justifies the name continued fraction. Note that cf (Γ) < 1. The usual notation for a lens space L(p, q), defined as p -surgery on the unknot, can recovered from the q continued fraction as follows. Write cf (Γ) = p, so that p > q 1 and (p, q) = 1. q Then PΓ = L(p, q). Remark Note that if Γ is a nonempty linear plumbing graph in normal form which is not a 0-weighted single vertex then detγ 0. We will make extensive use of this fact throughout this work without further references. A closed Seifert fibered manifold (see [18] and the references therein) can be described by its unnormalized Seifert invariants (g; b; (α 1, β 1 ),..., (α k, β k )) where g 0 is the genus of the base surface, b Z, α i > 1 and (α i, β i ) = 1. This data, (which is not unique), uniquely determines the manifold. When g = 0 a surgery description for such a manifold is depicted in Figure 1.1. Figure 1.1: A surgery description for the Seifert fibered manifold (0; b; (α 1, β 1 ),..., (α k, β k )). The following theorem is proved in [18].

15 1.1 Plumbed manifolds 14 Theorem Let Γ be the following star-shaped plumbing graph in normal form. a 1 1 a 1 n 1... b a 2 1 a 2 n 2... a k 1. a k n k... Then PΓ is a Seifert manifold with unnormalized Seifert invariants (0, b; (α 1, β 1 ),..., (α k, β k )) where The quantity α i β i = [a i 1,..., a i n i ]. e(y) := b k i=1 β i α i is called the Euler number of Y. It is easy to check that e(y) = cf (Γ) (1.1) where Γ is the plumbing graph in normal form associated to Y. Definition Let Γ 1 and Γ 2 be two linear plumbing graphs in normal form. a 1 a 2 a n Γ 1 :=... b 1 b 2 b m Γ 2 :=... Γ 1 and Γ 2 are said to be complementary if Γ 2 = Γ 1. Proposition With the notation of Definition 1.14 the following conditions are equivalent 1. Γ 1 and Γ 2 are complementary

16 1.1 Plumbed manifolds P( b m b 1 1 a 1 a n ) = S 1 S cf (Γ 1 ) + 1 cf (Γ 2 ) = 1 Proof. (1) (2). This can be checked directly using Theorem 1.9. A series of 1- blowdowns will turn the linear graph above in a 0-weighted single vertex. (2) (3). Consider the continued fraction of the graph representing S 1 S 2 with respect to the only 1-weighted vertex. We have 0 = det( b m b 1 1 a 1 a n ) det(γ 1 )det(γ 2 ) = 1 1 cf (Γ 1 ) 1 cf (Γ 2 ). The first equality above holds because for any plumbing graph we have b 1 ( PΓ) = b 0 (Γ). (3) (1). By the same formula used above we obtain det( b m b 1 1 a 1 a n ) = 0 After a 1-blowdown we obtain det( b m b a a n ) = 0 therefore this plumbing graph is not in normal form, which means that at least one weight among a 1 and b 1 is. Suppose, for instance, that a 1 =. If n = 1 it is easy to see that m = 1 as well, and b 1 = from which the conclusion follows. Therefore we may assume that n > 1. If m = 1 by blowing down the vertex whose weight is a we obtain b a a n... Again, this graph has vanishing determinant and therefore is not in normal form. If b 1 = 3 we blow down the vertex whose weight is b It follows easily that a 2 = and that n = 2. If b 1 < 3 then a 2 =, we blow down the vertex whose weight is a and we iterate the argument. This shows that (a 1,..., a n ) = (,..., ) and

17 1.1 Plumbed manifolds 16 that n = b 1 1. If m > 1 we claim that b 1 3. To see this, assume by contradiction that b 1 =. By blowing down the vertex whose weight is a we obtain b m b 2 0 a a n which, by 0-chain absorption becomes b m b 2 + a a 3 a n This last graph is in normal form which contradicts the fact that its determinant is zero. This proves the claim. Now the argument can be iterated. Each time we blow down a 1-vertex we obtain a new linear graph which has exactly one 1-vertex. By repeatedly blowing down 1-vertices we will eventually obtain the graph b m 1 a n since the determinant must vanish it is easy to verify that a n = b m = and that b m 1 a n P( ) = S 1 S 2. This proves (2) and, by Theorem 1.9 also (1). Infact, by induction on the number of blow up operations one can verify that each linear graph corresponds to a pair of complementary strings. This can be done by starting with the last graph we obtained above and then going backwards via blow ups. Remark Note that, strictly speaking, the definition of complementary linear graphs should involve an extra bit of data. In Definition 1.14 we implicitly fixed an initial vertex and a final one on each graph (as suggested by the indexing of the weights). Only in this way the condition Γ 2 = Γ 1 makes sense. It is useful to extend in the obvious way the notion of complementary linear graphs to that of complementary legs in a star-shaped plumbing graph. We also say that a pair of Seifert invariants are complementary if they correspond to complementary legs in the associated star-shaped plumbing graph in normal form. It follows by Proposition 1.15 that pairs of complementary legs correspond to pairs of Seifert invariants (α i, β i ) and

18 1.1 Plumbed manifolds 17 (α j, β j ) that satisfy β i α i + β j α j = 1 Note that, in general, this formula does not hold if we do not compute the Seifert invariants from the weights of a star-shaped plumbing graph in normal form as in Theorem The linear complexity of a tree Let Γ be a plumbing graph in normal form. Let lc(γ) be the cardinality of the smallest subset of vertices we need to remove from Γ in order to obtain a linear graph. We call lc(γ) the linear complexity of Γ and we set lc( ) = 1. We stress the fact that because of the uniqueness of the normal form of a plumbing graph it makes sense to talk about the linear complexity of a plumbed 3-manifold. Note that: lc(γ) = 0 if and only if PΓ is a lens space if PΓ is a Seifert manifold then lc(γ) = 1 lc(γ 1 Γ 2 ) = lc(γ 1 ) + lc(γ 2 ). Proposition Let Γ be a plumbing graph in normal form such that lc(γ) = 1 and for at least one choice of a vertex v Γ the graph Γ v is linear and negative definite. Then detγ = 0 cf Γ = 0. Proof. The proof follows directly from Definition 1.8 and we omit the details. In Chapter 2 we will deal mainly with plumbed 3-manifolds with lc(γ) = 1. A generic plumbing graph Γ with lc(γ) = 1 looks like the one shown below Such a graph is made of a distinguished vertex v and several linear components. These linear components are joined to v via a final vertex (on the left-hand side of the picture above) or via an internal vertex (right-hand side).

19 1.2 Arborescent links Arborescent links In this section we introduce arborescent links. We recall the fact that the branched double cover of an arborescent link is a plumbed manifold and that every plumbed manifold can be realized in this way. Given a plumbing graph Γ, we recall the fact that all arborescent links whose branched double cover is P(Γ) are related to each other by Conway mutation. Finally we observe that combining results from [16] and [1] it is possible to define a normal form for the plumbing graph of an arborescent link from looking at its branched double cover. An embedded plumbing graph is a plumbing graph Γ together with an embedding i : Γ R 2. Two embeddings are considered the same if they differ by an orientation preserving selfhomoerphism of R 2. To any embedded plumbing graph we can associate a surface obtained by plumbing together twisted bands according to i(γ). Figure 1.2: The plumbing operation. See Figure 1.2 for an example and [1] for a precise definition. On each band we need to specify several plumbing patches, between two plumbing patches the band may be twisted. This configuration is described by an embedded tree together with integral weights on the angular sectors of each vertex. It is easily seen, however, that by appropriately flipping the tangles determined by the plumbing patches we may assume that only one angular sector carries a non zero weight. (see Section 12.4 in [1], this is called the Weight Condition). Therefore given an embedded plumbing graph together with the choice of an angular sector on each vertex we can think of the weight of the vertex as the weight of the angular sector and associate to it a well defined arborescent link. See Figure 1.3 for a sufficiently general example.

20 1.2 Arborescent links Figure 1.3: An example of an embedded plumbing graph and a diagram of its associated arborescent link. Note that the weights are assigned to the angular sectors rather than to the vertices. 0-weights are omitted. It is shown in [1] that this construction gives an embedded surface S(Γ) (actually, a family) whose boundary is a well defined (unoriented) link L(Γ). An arborescent link is a link contructed in the above way. We denote the collection of all oriented links constructed from a given plumbing graph Γ with L(Γ). The following theorem can be proved using results from [7] (Chapter 6). Theorem Let Γ be an embedded plumbing graph, and let S(Γ) be a corresponding plumbed surface. The 4-manifold PΓ is orientation preserving diffeomorphic to the branched double cover of the 4-ball branched along a pushed-in copy of S. In particular PΓ is the branched double cover of S 3 branched along S(Γ) = L(Γ). Remark We will usually denote an embedded plumbing graph by Γ, without any reference to the embedding or the choices for the angular sectors. It will be clear by the context if the graph is to be considered embedded or not. Remark An embedding i : Γ R 2 does not depend on the weights of Γ. It determines a cyclic order on the edges adjacent to every vertex. In particular blowing downs, 0-chain absorptions and splittings can be performed on embedded plumbing graphs. Proposition ([1]) Let Γ be an embedded plumbing graph. Suppose that Γ is obtained from Γ via a sequence of (embedded) blowing downs, 0-chain absorptions and splittings. The links L(Γ) and L(Γ ) are isotopic. Corollary Let Γ be a (not necessarily embedded) plumbing graph. The collection of isotopy classes of links L(Γ) only depends on the (oriented) diffeomorphism type of PΓ. In particular each L L(Γ) can be realized by the following data

21 1.2 Arborescent links 20 the plumbing graph in normal form Γ such that PΓ = PΓ an embedding i : Γ R 2 a choice of an angular sector on each vertex of Γ Proposition Let Γ 1, Γ 2 be two plumbing graphs such that P(Γ 1 ) and P(Γ 2 ) are diffeomorphic. For any choice of embeddings for Γ 1 and Γ 2 and angular sectors for the weights, the corresponding links L(Γ 1 ) and L(Γ 2 ) are obtained from each other via a finite sequence of Conway mutations. Proof. Let L i L(Γ i ), where i = 1, 2, be the two arborescent links which are determined by the choice of embeddings for Γ 1 and Γ 2. The projections of these links can be chosen so that they correspond to the normal form of Γ 1 and Γ 2. This is done by performing embedded elementary moves on the plumbing graphs which, by Proposition 1.21 correspond to isotopies on L 1 and L 2. Now it suffices to show that any two embeddings of the same plumbing graph produce links which are related to each other by a series of Conway mutations. It is enough to show that a single transposition between two adjacent subtrees corresponds to a Conway mutation. This fact is schematically illustrated in Figure 1.4.

22 1.2 Arborescent links 21 a. b. c. Figure 1.4: Exchanging two edges on an embedded plumbing graph via Conway mutation Montesinos links We conclude this chapter introducing Montesinos links. Definition Let Γ be a star-shaped plumbing graph in normal form. Any arborescent link L(Γ) is called a Montesinos link. In this specialized context (star-shaped plumbing graphs, Seifert manifolds, Montesinos links) the relation between plumbed manifolds and arborescent links can be refined in order to provide a classification theorem. The following result is due to Bonahon and Siebenmann [1]. Theorem Two Montesinos links L(Γ 1 ) and L(Γ 2 ) are isotopic if and only if Γ 1 can be obtained from Γ 2 via a sequence of embedded elementary moves and their inverses.

23 1.2 Arborescent links 22 Note that the embedding of a star-shaped plumbing graph determines, and is uniquely determined by, a choice of ordering of the legs up to cyclic permutation and reversal of order. Therefore a Montesinos link can be specified by the unnormalized Seifert invariants of its branched double cover (omitting the genus of the base surface) L := (b, (α 1, β 1 ),..., (α k, β k )) with the understanding that the order of the pairs (α i, β i ) has been fixed.

24 Chapter 2 Homology cobordisms and slice links In this chapter we state our main results on plumbed manifolds and arborescent links. In Section 2.1 we generalize to multicomponent links and make explicit an obstruction to the sliceness of a knot based on homology cobordism properties of cyclic branched covers. In Section 2.2 we explain how to construct arborescent slice links and plumbed manifolds with the corresponding cobordism property. In Section 2.3 we state our main results. We introduce a family of plumbed manifolds which, up to orientation reversal, includes all Seifert fibered manifolds over the 2-sphere. We prove that the only nullcobordant manifolds in this family are those resulting from the construction introduced in Section 2.2. In Section 2.5 we state the main corollary of our result which is a mutated version of the slice-ribbon conjecture for an infinite family of multicomponent arborescent links that includes all Montesinos links. In Section 2.6 we speculate on the problem of defining a group structure for cobordism classes of 3-manifolds with b 1 0. We conclude the chapter in Section 2.7, where we show how to use our construction to produce some, apparently new, examples of integral homology spheres bounding homology balls. These examples are obtained as 1 -surgeries on certain 2-bridge knots. n 2.1 Slice links and branched covers Two closed, oriented 3-manifolds Y 1,Y 2 are rational homology cobordant (or QHcobordant) if there exists a smooth compact 4-manifold W such that: W = Y 1 Y 2 both inclusions Y i W induce isomorphisms H (Y i ; Q) = H (W; Q). The set of oriented rational homology spheres up to rational homology cobordism is an abelian group with the operation induced by connected sum. We denote this group by Θ 3 Q. The zero element is given by (the equivalence class of) S3. Note that Y is QHcobordant to S 3 if and only if it bounds a smooth rational homology ball. It is well known that if a rational homology sphere is obtained as the branched double cover along a slice knot then it bounds a rational homology ball. In the next proposition 23

25 2.1 Slice links and branched covers 24 we make an analogous observation concerning branched double covers along slice links with more than one component. Proposition 2.1. Let L S 3 be a link. Let S D 4 be a properly embedded smooth surface without closed components such that S = L. Let W be the double cover of D 4 branched along S. Assume that b 1 ( W) χ(s) 1. Then b 1 (W) = χ(s) 1 and b 2 (W) = b 3 (W) = 0. In particular, if b 1 ( W) > 0 we have an isomorphism H (W; Q) = H ( χ(s) 1 i=1 S 1 D 3 ; Q). Proof. As shown in [12] we have a long exact sequence H i (D 4, S S 3 ) H i (W, W) H i (D 4, S 3 ) H i 1 (D 4, S S 3 )... from which we obtain an isomorphism H 1 (D 4, S S 3 ) = H 1 (W, W). It follows from the exact sequence of the pair that H 1 (D 4, S S 3 ) = 0. We conclude that 0 = H 1 (W, W) = H 3 (W). From the exact sequence of the pair (W, W) with rational coefficients we get H 1 ( W) H 1 (W) 0. We obtain b 1 (W) b 1 ( W) χ(s) 1. Since χ(w) = 2χ(B 4 ) χ(s) = 2 χ(s) 1 b 1 (W) + b 2 (W) = 2 χ(s) we see that b 1 (W) = χ(s) 1 and b 2 (W) = 0. Corollary 2.2. Let L be a slice link with n components (n > 1). Let W be the branched double cover of the four-ball branched along a collection of slicing discs for L. We have an isomorphism H (W; Q) = H ( n 1 i=1 S1 D 3 ; Q) Proof. It is well known that b 1 ( W) = L 1 (see for instance [9]). Here L denotes the number of components of the link L. Then, we may apply Proposition 2.1.

26 2.1 Slice links and branched covers 25 Motivated by Proposition 2.1 we investigate QH-cobordisms of plumbed 3-manifolds with b 1 1. Note that if a 3-manifold Y bounds a QH n S 1 D 3 then b 1 (Y) equals the number of S 1 D 3 summands. Proposition 2.3. Let Y be a connected 3-manifold with b 1 (Y) = n. Suppose that Y bounds smooth 4-manifolds X and W with the following properties: X is simply connected, negative semidefinite and rkq X = b 2 (X) n H (W; Q) = H ( n i=1 S1 D 3 ; Q) Then there exists a morphism of integral lattices ((H 2 (X); Z), Q X ) (Z b 2(X) n, Id). In particular for every definite sublattice (G, Q G ) (H 2 (X), Q X ) whose rank is b 2 (X) n we obtain an embedding of integral lattices (G, Q G ) (Z b 2(X) n, Id). Proof. Consider the smooth 4-manifold X := X Y W. The Mayer-Vietoris exact sequence with integral coefficients reads H 3 (X ) H 2 (Y) H 2 (X) H 2 (W) H 2 (X ) H 1 (Y) H 1 (W) H 1 (X ) 0 Note that b 1 (Y) = b 1 (W), moreover the map H 1 (Y; Q) H 1 (W; Q) is an isomorphism. It follows that b 1 (X ) = 0. The group H 2 (W) is finite. Note that b 3 (X ) = 0. If we consider the above exact sequence with rational coefficients we obtain 0 H 2 (Y; Q) H 2 (X; Q) H 2 (X ; Q) 0 therefore b 2 (X ) = b 2 (X) b 2 (Y) = b 2 (X) n. Now note that, by Novikov additivity, σ(x ) = σ(x). This shows that X is a smooth, closed negative definite 4-manifold. By Donaldson s diagonalization theorem its intersection form is equivalent to the standard negative definite form on Z b 2(X ). The inclusion X X induces the desired morphism of integral lattices. The last assertion follows easily. The map ϕ : (G, Q G ) (Z b 2(X) n, Id)

27 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class 26 preserves the intersection form. Since Q G is negative definite, ϕ must be injective and is therefore an embeddingof integral lattices. 2.2 Building plumbed 3-manifolds with prescribed QHcobordism class Recall that a rooted plumbing graph (Γ, v) is a plumbing graph with a distinguished vertex. In particular, a rooted plumbing graph is necessarly nonempty. Definition 2.4. Let (Γ 1, v 1 ) and (Γ 2, v 2 ) be two rooted plumbing graphs. Let Γ be the plumbing graph obtained from Γ 1 Γ 2 by identifing the two distinguished vertices and taking the sum of the corresponding weights. together Γ 1 and Γ 2 along v 1 and v 2 and we write Γ := Γ 1 v 1,v 2 Γ 2. We say that Γ is obtained by joining The following proposition follows immediately from Proposition 1.8 Proposition 2.5. With the above notation we have cf (Γ 1 v 1,v 2 Γ 2 ) = cf (Γ 1 ) + cf (Γ 2 ) provided that the continued fractions on the right are computed with respect to the vertices v 1 and v 2, and the continued fraction on the left is computed with respect to the vertex resulting from joining v 1 and v 2. Lemma 2.6. Let W be a connected 4-dimensional handlebody without 3-handles. If H ( W; Q) = H (S 3 ; Q) then H 1 (W; Q) = 0. In particular, if W is built using a single 1-handle h 1, and a single 2-handle h 2 then the algebraic intersection of these handles does not vanish. Proof. The homology exact sequence of the pair (W, W) with rational coefficients reads H 1 ( W) H 1 (W) H 1 (W, W) 0. Since H 1 ( W) = 0 and by Lefschetz duality H 1 (W, W) = H 3 (W) = 0 the conclusion follows. If there are only two handles h 1 and h 2, the attaching sphere of h 2 must have

28 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class 27 nonzero intersection number with the belt sphere of h 1, otherwise h 1 would represent a non trivial element in H 1 (W). The following lemma is an immediate consequence of the splitting move. Lemma 2.7. Let (a 1,..., a n ) and (b 1,..., b m ) be strings (where each coefficient is ). The 3-manifold described by the plumbing graph b 1 b m... a n a is a connected sum of two lens spaces. Remark 2.8. If in the previous lemma we choose two complementary strings the plumbing graph depicted above, with the 0-weighted vertex removed, represents S 1 S 2. However, not every linear plumbing graph that represents S 1 S 2 has this form. Apart from some obvious examples like ; ; there are also examples where there is a 1-weighted internal vertex. For instance ; These examples show that the assumption on the weights in Lemma 2.7 is necessary. This follows from the fact that removing the central vertex in the two graphs above one obtains a plumbing graph that represents S 1 S 2 S 1 S 2, instead of a rational homology sphere. Proposition 2.9. Let (Γ, v) be a rooted plumbing graph such that P(Γ) = S 1 S 2 and P(Γ \ {v}) is a rational homology sphere. Let (Γ, v ) be any rooted plumbing graph. Then for any link L 0 L(Γ ) there exist a link L 1 L(Γ v,v Γ) and a properly embedded smooth surface S S 3 I such that

29 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class S S 3 {0} = L 0 2. S S 3 {1} = L 1 3. χ(s) = 0 4. the branched double cover of S 3 I along S realizes a QH-cobordism between P(Γ ) and P(Γ v,v Γ) In particular b 1 ( P(Γ )) = b 1 ( P(Γ v,v Γ)) and these manifolds are QH-cobordant. Proof. We can choose a diagram for L 0 so that the band corresponding to the distinguished vertex of Γ and all the bands plumbed to it look like those depicted on the right in Figure 2.1(a). On the left of Figure 2.1(a) we have a surgery description for the double cover of S 3 branched along L 0 (see also [11], figure 3). Let S S 3 I be the obvious surface consisting of annuli and a single disk such that S S 3 {0} = L 0 and S S 3 {1} = L 0. We may think of L 0 also as L 0 2 (here n denotes the trivial link with n components). Taking the branched double cover of S 3 I along S we obtain a smooth 4-manifold W whose boundary is PΓ P(Γ Γ). We are now in Figure 2.1(b). On the left we have a surgery diagram for the boundary of the 4-manifold obtained from PΓ by attaching a 4-dimensional 1-handle. corresponds to a 2-dimensional 0-handle attachment on the right. Note that P(Γ Γ) = PΓ PΓ = PΓ S 1 S 2. This By [10] the only link whose branched double cover is S 1 S 2 is the trivial link with two components (see also [11] pag. 21). In particular any embedding in the plane of the plumbing graph Γ is associated to an arborescent diagram of the trivial link with two components. Therefore, choosing an embedding of Γ in the plane we can draw a corresponding diagram for L 0 2. This is done on the right of Figure 2.1(c). In other words the surface whose boundary is given by the links on the right in 2.1(b) and 2.1(c) is just a product L 0 2 [1, 2]. On the left the corresponding surgery diagrams describe the same 3-manifold.

30 Building plumbed 3-manifolds with prescribed QH-cobordism class 29 a. k k b. k k c. k h k h d. k 0 h k h e. k+h k+h Figure 2.1: On the right a movie description for the properly embedded smooth surface S S 3 I. On the left each diagram is a surgery description for the 3-manifold obtained as the branched double cover along the corresponding link on the right.

31 Building plumbed 3-manifolds with prescribed QH-cobordism class 30 If we attach a 0-framed 2-handle to W as shown on the left in Figure 2.1(d) we obtain a 4-manifold W whose boundary is now Γ P(Γ v,v Γ). Moreover this handle attachment can be made equivariant with respect to the involution that corresponds to the surface S. This operation corresponds to attaching a band to the surface S performed on the link S S 3 {2}, see Figure 2.1(d) on the right. We have constructed a cobordism W between PΓ and P(Γ v,v Γ) which consists of one 1-handle and one 2-handle. This cobordism is obtained as the branched double cover of a surface S S 3 I. Up to isotopy we may draw a different diagram for the bottom boundary of S, in this way we obtain the diagram depicted on the right of Figure 2.1(e). This isotopy corresponds to a 0-chain absorption on the surgery diagram on the left. In order to prove that W is in fact a QH-cobordism it suffices to check that the algebraic intersection between the attaching sphere of the 2-handle and the belt sphere of the 1-handle does not vanish. Let us write α for the attaching sphere of the 2-handle. The first homology group of PΓ S 1 S 2 is Q b 1( PΓ ) Q. Our algebraic intersection number is non zero if and only if α represents a non trivial element when projected into H 1 (S 1 S 2 ). Note that in H 1 ( PΓ S 1 S 2 ) the curve α is homologous to the pair of curves α 1 and α 2 shown in Figure 2.2. k 0 h k 0 0 h Figure 2.2: The thick curve on the leftmost diagram is homologous to the sum of the two thick curves on the rightmost diagram. This means that the projection of α in H 1 (S 1 S 2 ) is equivalent to α 2. The fact that α 2 is a nontrivial element in H 1 (S 1 S 2 ) follows immediately from our hypotheses on (Γ, v). To see this, let L be the link that gives a surgery description for S 1 S 2 in Figure 2.1(c) (the rightmost link in the leftmost picture). Applying the splitting move on the link α 2 L we see that the 3-manifold described by this link is precisely P(Γ \ {v}), which by our assumption is a rational homology sphere. This fact ensures that α 2 represents a non trivial element in H 1 (S 1 S 2 ; Q). It follows that b 1 ( PΓ ) = b 1 ( P(Γ v,v Γ)) and that W is a QH-cobordism.

32 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class 31 It is clear from our construction that the cobordism W has been built as the branched double cover of a properly embedded smooth surface S S 3 I. Finally note that 0 = χ(w ) = 2χ(S 3 ) χ(s) χ(s) = 0. Remark The simplest way to use the above Proposition is to choose (Γ, v) as any graph like the ones in Remark 2.8, the vertex v being one of those whose weight is 1. Remark The 2-handle attachment used in Proposition 2.9 can also be described in terms of plumbing graphs as follows. We start with P(Γ Γ) which has the following description w(v ) 1 a 1 a n... b 1 b m... Where, for simplicity, we have choosen Γ as in Lemma 2.7. The 2-handle then appears as an additional vertex as shown below. w(v ) 0 1 a 1 a n... b 1 b m... This last level of the cobordism can be described also by the following plumbing graph, using the 0-chain absorption move. w(v ) 1 a 1 a n... b 1 b m...

33 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class 32 Example Let (a 1,..., a n ) and (b 1,..., b m ) be two complementary strings. The plumbing graph associated to the string (a n,..., a 1, 1, b 1,..., b m ) represents S 1 S 2. By the previous proposition all lens spaces associated to strings of the form (a n,..., a 1,, b 1,..., b m ) are QH-cobordant to S 3. In fact, the corresponding plumbing graph is obtained by joining together a 1-weighted vertex and a graph as in Remark 2.7. Example Choose strings (a i n i,..., a i 1, 1, bi 1,..., bi m i ), where i = 1,..., k, as in the previous example. Consider the plumbed manifold described by the following star-shaped plumbing graph a 1 1 a 1 n 1... b 1 1 b 1 m 1... k.. a k 1 a k n k... b k 1 b k m k... This graph can be obtained by starting with a 0-weighted single vertex and then joining k graphs of the type described in Remark 2.7 along their 1-vertices. More precisely one should apply the joining operation k times. By applyng Proposition 2.9 at each step we see that such a manifold is QH-cobordant to S 1 S 2 and thus it bounds a QH S 1 D 3. In Section 2.3 we will see that these are the only Seifert manifolds over the 2-sphere with this property Elementary building blocks In the previous example we have used the graph Γ 1 := a n a 1 1 b 1 b m......

34 2.2 Building plumbed 3-manifolds with prescribed QH-cobordism class 33 as a building block for constructing rational homology cobordisms of 3-manifolds. This is somehow the simplest way to use Proposition 2.9. The process can be iterated by constructing more complicated pieces to be used as building blocks. Keeping in mind that we are interested in plumbed manifolds with lc = 1 we may introduce three more building blocks. The graph Γ 1 can be slightly modified obtaining a n a 1 b 1 b m Γ 2 := Another building block can be obtained starting with 1 n... where n 1 is the length of the -chain (n 2). This is just a special case of Γ 1. Now we join this graph with Γ 1 along the vertices of weight n and 1. We obtain our third building block 1 n 1 Γ 3 :=... a 1 a n... b 1 b m... Note that PΓ 3 = S 1 S 2. A fourth building block can be constructed as follows. We start with 1 and then we attach to the final vertices of this graph two linear graphs like Γ 1. We obtain a n a 1 a 1 a n Γ 4 := b m b 1 b 1 b m Note that this last graph does not represent S 1 S 2 since its normal form can be obtained

35 2.3 Main theorem 34 by blowing down the 1-vertex. Each of the four building blocks we have introduced have a distinguished 1-weighted vertex. From now on we will implicitly consider each of these graphs as a rooted plumbing graph where the prefered vertex is the one whose weight is 1. Definition The four families of rooted plumbing graphs introduced above will be called building blocks of the first, second, third and fourth type, respectively. Of course, using Proposition 2.9, one can construct many examples of plumbed 3-manifolds with arbitrarly high linear complexity that are QH-cobordant to S 1 S 2. A simple example is given by the following graph 3 whose linear complexity is 2. The following proposition is an immediate consequence of Proposition 2.9. Proposition Let Γ be a plumbing graph obtained by joining together two or more building blocks of any type along their 1-vertices. Then 1. Γ is in normal form 2. lc(γ) = 1 3. PΓ bounds a QH S 1 D 3. Our main result, Theorem 2.16, should be thought of as a converse of this last proposition. 2.3 Main theorem Before we state our main result we introduce some terminology. Let Γ be a plumbing graph in normal form such that lc(γ) = 1. Choose v Γ such that Γ := Γ \ {v} is linear. The linear graph Γ is a disjoint union of connected linear graphs Γ 1,..., Γ k. We call Γ i a final leg or an internal leg according to wether v is linked to a final vertex of Γ i or an internal one. We indicate with i(γ, v) and f (Γ, v) the number of internal (resp. final)