When Closed Graph Manifolds are Finitely Covered by Surface Bundles Over S 1

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1 Acta Mathematica Sinica, English Series 1999, Jan, Vol15, No1, p When Closed Graph Manifolds are Finitely Covered by Srface Bndles Over S 1 Yan Wang Fengchn Y Department of Mathematics, Peking University, Beijing P R China yanw@sxx0mathpkedcn Abstract The problem of deciding whether a graph manifold is finitely covered by a srface bndle over the circle is discssed in this paper A necessary and sfficient condition in term of the soltions of a certain matrix eqation is obtained, as well as a necessary condition which is easy to compte Or reslts sharpen and extend the earlier reslts of J Lecke-Y W, W Nemann, and S Wang-F Y in this topic Keywords Srface bndle, Covering, Graph manifolds 1991MR Sbject Classification 57N10, 57M10, 15A18 A classical fact is that Seifert manifolds with non-empty bondaries are finitely covered by srface bndles over S 1 And closed Seifert manifolds are covered by srface bndles over S 1 if and only if its Eler s nmber is zero In the past several years, researchers have proved that some closed graph manifolds are not covered by srface bndles over S 1 ([1,2]) and graph manifolds with non-empty bondaries are finitely covered by srface bndles over S 1 ([3]) In this paper we discss the case when a closed graph manifold is finitely covered by a srface bndle over S 1 For a given closed graph manifold M, there are two associated matrixes B(M) and W (M) (for details see the next section), where B(M) is defined in [4] and W (M) in[3] We have two reslts The first reslt is Theorem 21, which is a necessary condition for M to be covered by srface bndles This condition can be read directly from the entries of the matrices B(M) andw (M), therefore this reslt is very sefl in practice The second reslt is Theorem 31, which is a necessary and sfficient condition for M to be covered by srface bndles The condition is that a system of eqations, whose coefficients are related with B(M) and W (M) in qite a complicated way, has a certain soltion Theoretically Theory 31 is a Received Jne 12, 1998, Revised Jly 25, 1998, Accepted September 9, 1998 Spported in part by NSFC

2 12 Yan Wang & Fengchn Y fine reslt Bt it is not easy to compte in practice, in particlar, it is hard to claim that the system does not have a reqired soltion So Theorem 21 cannot be replaced by Theorem 31 We recognized that or Theorem 31 coincided sbstantially with the main reslt in [2] However we do not need the condition no self-pasting as specified in the main reslt in [2] 1 Definitions and Propositions Definition 11 A compact irredcible -irredcible orientable 3-manifold M is a graph manifold if each component of M τ is a Seifert manifold, where τ is the canonical decomposition tori of Jaco-Shalen and of Johanson Note We can refer to [5] or [6] for the definition of a Seifert manifold Definition 12 A graph manifold is trivial if and only if it is covered by srface S 1 or some tors bndle over S 1 We may assme in this article that the graph manifolds concerned are all closed, oriented, and non-trivial Sppose M is a non-trivial graph manifold, then it has a decomposing srface S (inclding tors and Klein bottle) ([4, Section 3]) We se N(S) to denote a reglar neighborhood of S, and η(s) to denote the interior of N(S) For a given graph manifold M, we call each component M v of M S a vertex manifold Define an associated graph Γ(M) asbelow: To each component M i of M η(s), a vertex v i is assigned, and to each component S j of S an edge e j is assigned, so that (1) if S j is a tors, and N(s j ) has one component in each of M i and M k (i may be eqal to k), then e j has endpoints on v i and v k (2) if S j is a Klein bottle, and N(s j )isinm i,thene j has both endpoints on v i Consider N = T I, a trivial I-bndle over a tors T Sppose N has been given an S 1 - fibration strctre ξ Choose a fiber α i on T i for i =0, 1, and let α 1 be the crve on T 0which is isotopic to α 1 in N We define the fiber intersection nmber on T as (T )= ξ (T )= α 0 α 1, where α 0 α 1 is the minimm intersection nmber of the two crves on T 0 Now sppose N is a twisted I-bndle over a Klein bottle K Then N is a single tors Up to isotopy, N has exactly two Seifert fibered strctres We denote by C 1 acrveon N that is a fiber of the fibration whose orbifold is a Möbis band, and denote by C 2 afiberon N of the other fibration They form a coordinate system on N Given an arbitrary S 1 -fibration ξ on N, afiberα of ξ represents a niqe element ac 1 + bc 2 in H 1 ( N), We define the fibre intersection nmber on the Klein bottle K to be (K) = 4ab Definition 13 AtorsT is framed, if T is oriented and two ordered oriented simple closed circles α, β, which intersect transversely exactly once, are chosen so that the prodct of their orientations prodces the orientation of T Sch a framed tors is also sometimes denoted as

3 When Closed Graph Manifolds are Finitely Covered by Srface Bndles Over S 1 13 T (α, β) Sppose P : M F is an oriented Seifert manifold, where the orbit srface F is of gens g and has h>0 bondary components and M has k singlar fibers M is framed, if (1) a section S = F int D i of M int N i is chosen and S is oriented, where N is a fibered reglar neighborhood of the singlar fibers And D is a reglar neighborhood of P (N) (2) eqip each tors bondary component T with a framing T (α, β), where α is an oriented bondary component of F and β is an oriented fiber S 1 (3) the orientation of T is the indced orientation from M An orientable graph manifold M is framed, if (a) each vertex manifold is framed and the orientation on each M v coincides with the restriction of the orientation of M (b) the graph Γ(M) is oriented Note (1) We can refer to [7] for the definitions above (2) In the definitions above we exclde the Klein bottle, becase its fibre intersection nmber is given natrally If we pt an orientation on e for each e Γ(M), then e determines ( a homeomorphism ) g e : T e (α e,β e ) T e (ᾱ e, β p e q e e ), and it determines niqely a 2 by 2 matrix, defined r e s ( ) ( ) ( ) e α e p e q e ᾱ e as g e = Where T e and T e are tori in M i and M k corresponding β e r e s e β e to the beginning and the end of e respectively It can be easily seen that r e is the fibre intersection nmber defined above, so r e 0and q e r e p e s e =1 Consider a covering map φ: M M, thenφ 1 (S) = S is a decomposing srface for M Therefore φ indces a map on the graphs φ:γ( M) Γ(M) It is very easily defined A vertex ṽ p is mapped to v i if the corresponding component M p in M η( S) ismappedtom i,andan edge ẽ q is mapped to e j if the corresponding srface s q covers s j For each vertex v i in Γ(M), define I i = {p φ(ṽ p )=v i }Wese i to denote the nmber of elements in I i Inotherwords, i is the nmber of components in M η( S) that cover the component M i in M η(s) We define a weight for each vertex or edge of Γ(M) as follows : If v i is a vertex of Γ(M) corresponding to a component M i of M η(s), let the weight χ i = χ(v i ) be the Eler characteristic of the orbifold of M i, If e is an edge corresponding to a srface S, let the weight γ(e) be the fibre intersection nmber (s) We define a symmetry matrix B(M) =(b ij ) n n and a diagonal matrix W (M) =diag(w 1,, w n ) by defining b ij = 1, w i = p e γ e:v e χ i χ j γ i v j e:v e χ 2 i i Where e : v i denotes the edges from v i and each edge e: v i v i shold be conted twice Definition 14 ([3, Definition 03]) An embedded srface S in a graph manifold M is horizontal, if S transverses S 1 -fibers of all vertex manifolds of M

4 14 Yan Wang & Fengchn Y [3, Lemma 05] has proved that if a graph manifold M admits a horizontal embedded srface S, thenm is covered by a srface bndle over S 1 Lemma 11 ([3, Theorem 15]) Sppose M is a closed framed graph manifold, Then M admits a horizontal srface S if the following eqation holds 0 (B W ) =, λ n 1 0 λ n 0 λ n 0 Note In [3, Theorem 15], the vector on the right of = is non-zero, it has an ndetermined *, that is (0,,0, ) T That is becase the manifolds discssed in [3] are with non-empty bondaries and the bondaries is assmed in the n vertex manifold It is the same with the following conclsions Lemma 12 ([3, Theorem 23]) Sppose φ: M M is a framing, preserving, covering of degree d between graph manifolds Then the B-matrixes and W-matrixes of M and M satisfy the following eqations: d B =(b ij X ij ) n n, B =(b ij ) Where X ij is a i by j sbmatrix sch that the row (colmn) sm is the constant j ( i ),if b ij 0 d W =diag( 1 w 1,, 1 w 1,, n w n,, n w n ), W =diag(w 1,,w n ) 1 n Proof Similar to the proof in Theorem 23 (see [3]), bt in this article, by replacing the compact srface F (see [3]) with the orbifold, we can easily arrive at the conclsion It is the same with the other lemmas derived from [3] 2 A Comptable Necessary Condition Let M be a closed graph manifold, B = B(M), W = W (M) =diag(w 1,,w n )bethebmatrix and W -matrix of M Let W =min( w 1,, w n ), (21) n B = max b ij, (22) i=1,2,,n j=1 Bx B =max x 0 x (23)

5 When Closed Graph Manifolds are Finitely Covered by Srface Bndles Over S 1 15 Where B is the maximm line sm norm of B Let ρ(b) be the spectral radis of B, that is the maximm eigen modls of B, thenρ(b) B Becase B = B T,wehave B = ρ(b) B Theorem 21 If the closed graph manifold M is finitely covered by a srface bndle over S 1, then either W (M) < B(M) or w 1 = w 2 = = w n = ± B(M) Proof If M is covered by a srface bndle over S 1, then by Lemma 11, we have B and W, sch that 0 ( B W ) =, (24) 0 λ 2 0 By Lemma 12 and the following Lemma 21, we can assme 1 = 2 = = n =, then From (24), we get d W = diag(w 1,,w }{{ 1,,w } n,,w n ) (25) B = W = w 1 w m (26) Then w 1 w m = B B (27) Clearly W w 1 w m (28) Becase (,, ) T 0, we have W B B (29) We can easily see that and we get W = d W, B = d B, (210) W B (211)

6 16 Yan Wang & Fengchn Y If W = B, then (27 29) are still valid, that is B = B (212) So (,, ) T is an eigenvector to ±ρ( B) Bt w 1 B = w m = ± W, λ i 0, (213) we get w 1 = w 2 = = w n = ± B (214) Lemma 21 Sppose B 1, W1 satisfy the eqations in Lemma 11 and Lemma 12, then we can find B, W which satisfy Lemma 11 and Lemma 12, and 1 = 2 = = n Proof Let Sppose d 1 W1 = diag( 1w 1,, 1w 1,, nw n,, nw n ) n 1 =[ 1,, n], that is the least mltiple of i s Let k i = Sppose i d 1 B1 =(( B 1 ) ij ), where ( B 1 ) ij is a i by j matrix Let d B =(( B) ij ), where ( B) ij =( B 1 ) ij is a by matrix; k i k j (215) d W =diag(w 1,,w }{{ 1,w } 2,,w 2,,w n,,w n ) (216) If ( B 1 W 1 ) =0,

7 When Closed Graph Manifolds are Finitely Covered by Srface Bndles Over S 1 17 and λ 2 0, we can get easily ( B W ) Where the nmber of λ i is k i λ 2 λ 2 λ ṃ =0 (217) Note A B =(a ij B) is the tensor prodct of matrix 3 A Necessary and Sfficient Condition Lemma 31 ([3, Lemma 31]) If there is a symmetric n by n matrix C =(c ij ) n n with rational entries and n non-zero rational nmbers,,λ n, sch that they satisfy the eqation 0 (C W ) =, (31) λ n 0 c ij b ij, then M can be finitely covered by a srface bndle over S 1 WhereM is a closed graph manifold, B =(b ij ) n n and W =diag(w 1,,w n ) is the B-matrix and W-matrix of M respectively In this part, we will see that for a closed graph manifold M, it is also a necessary condition for its being covered by a srface bndle over S 1 that B(M) andw (M) satisfy the eqation (31) Lemma 32 ([8, Theorem 511]) Let I be a finite set of indices, I = {1, 2,,n} Foreach i I, lets i be a sbset of a set S A necessary and sfficient condition for the existence of distinct representatives x i, i =1,,n, x i S i,x i x j,wheni j is condition C: For every k =1,,n and choice of k distinct indices i 1,,i k, the sbsets S i1,,s ik contain between them at least k distinct elements Lemma 33 Let X =(x ij ) be a matrix with non-negative rational entries with the row (colmn) sm the constant, then it has a decomposition (X) = σ s k σ P σ, k σ 0, (32)

8 18 Yan Wang & Fengchn Y where k σ is a non-negative rational nmber and S is the permtation grop of elements, σ S is a permtation, P σ is the permtation matrix determined by σ, that is { 1, if σ(i) =j, (P σ ) ij = 0, otherwise (33) Clearly σ s k σ = Proof we can refer to [8, theorem 519] for the conclsion Lemma 34 Let B =(b ij ) n n and W = diag(w 1,,w n ) be the B-matrix and a W-matrix of a graph manifold M, then for any given symmetry matrix with non-negative rational entries like (b ij X ij ) n n,wherex ij is a by matrix with non-negative entries with row (colmn)sm, there is a d-fold finite covering (not necessarily reglar) φ: M M, sch that db( M) =(b ij X ij ) n n, dw ( M) =diag(w 1,,w }{{ 1,,w } n,,w n ) (34) where B( M) and W ( M) are matrix to M Proof For any positive integer d 1 and each vertex manifold M i, there is a covering map q i : M i M i of degree d 1, sch that the restriction of q i on each bondary component onto its image is a homeomorphism Clearly all those q i : M i M i are matched well to give a covering P : M M For each v i Γ(M), there is a niqe v i Γ(M )overv i For each edge e in Γ(M), there are d 1 edges ẽ over e and clearly χ i = d 1 χ i and γ e = γ e,sowehave d 1 b ij = b ij (35) By Lemma 33, we can sppose X ij = k ij (σ)p σ Now choose the degree d 1 by the following eqation k ij (σ) = q ij(σ) p ij (σ) = l ij(σ), (36) d 1 where p ij (σ) andq ij (σ) are coprime and d 1 is the least mltiple of all p ij (σ) s Becase so k ij(σ) =1, σ s l ij (σ) =d 1 (37) σ s Now we can constrct a degree covering P : Γ Γ =Γ(M ) (38),

9 When Closed Graph Manifolds are Finitely Covered by Srface Bndles Over S 1 19 as below : Pt vertices (v i, 1), (v i, 2),,(v i,)overv i For each edge e: v i v j of Γ(M), divide the d 1 edges P1 1 (e)(p 1 :Γ Γ) of Γ connecting v i and v j into s =! grops We mark these grop as σ, τ, etc, σ, τ s There are l ij (σ) edgesintheσ grop We lift l ij (σ) edges of P1 1 (e) starting from (v i,k) sch that each edge in the σ grop ends at (v j,σ(k)) Let P : M M be the associated covering with Γ( M) = Γ and the restriction P on each vertex manifold of M is a homeomorphism Then define P = P P, Now we order the vertices of Γ by ṽ (i 1)+l =(v i,l),i =1,,n, l =1,, For each edge e connecting v i and v j, there are l ij (σ) σ(k)=l σ s edges connecting (v i,k)and(v j,l) Note that l ij (σ) = d 1 (X ij) kl σ(k)=l σ s 1 Let χ(e) = γ e χ iχ j (e : v i v j ), then χ(ẽ) = χ(e) d 2 1 So b(i 1)+k,(j 1)+l = that is We have proved the lemma e:v i v j σ(k)=l = 1 d 1 l ij (σ) χ(e) d 2 1 e:v i v j χ(e) (X ij ) kl = bij d (X ij) kl, (39) db( M) =(b ij X ij ) n n (310) Theorem 31 Sppose M is a graph manifold B =(b ij ) n n and W =diag(w 1,,w n ) be the B-matrix and a W-matrix of M Then M is covered by a srface bndle over S 1 if and only if there is a symmetric matrix with non-negative entries B =(b ij X ij ) n n, W = diag( 1 w 1,, 1 w 1, 2 w 2,, 2 w 2,, n w n,, n w n ) 1 n with rational entries, where X ij is a non-negative i j matrix with row (colmn) sm the constant j ( i ), which satisfy the eqation 0 ( B W ) =, (311) n k=1 2 λ n λ k 0

10 20 Yan Wang & Fengchn Y Proof On one hand, similar to Lemma 21 we can find B 1 and W 1 sch that 0 ( B 1 W 1 ) =, λ n 0 λ n 0 (312) has a soltion and 1 = 2 = = n = By Lemma 34, there is a d-fold covering φ: M 1 M, sch that db( M 1 )= B 1,dW( M 1 )= W 1 (313) Then M is covered by a srface bndle over S 1, by Lemma 11 On the other hand, if M is covered by a srface bndle over S 1, then there is a reglar covering φ 1 : M 1 M, sch that M 1 has an embedded horizontal srface and there is a soltion to ( B 1 W 1 ) =0, (314) λ 2 0 Where B 1 and W 1 are the B-matrix and W -matrix of M1 respectively Note that Similar to the proof of [3, Lemma 31], we can get a two-fold covering M M 1, sch ( B W ) λ 2m = λ 2 λ 2m 0 has a soltion Where 1 = 2 = = 2m =2 0 0, So in Theorem 31 we can only consider the case 1 = = n = (315) Acknowledgement help We are gratefl to Professor S C Wang and Dr H W Sn for their References [1] J Lecke, Y Q W Relative Eler nmber and finite covers of graph manifolds Proceedings of the Georgia International Topology Conference, AMS/AP, 1997, 12: [2] W D Nemann Commensrability and virtal fibration for graph manifolds Topology, 1997, (2): [3] S C Wang, F C Y Graph manifolds with non-empty bondaries are finitely covered by srface bndles Math Proc Camb Phil Soc, 1997, 122: [4] S C Wang, Y Q W Covering invariant of graph manifolds and cohopficity of 3-manifold grops Proc London Math Soc, 1994, 68: [5] J Hempel 3-manifolds Annals of Mathematics Stdies 86, Princeton University Press, 1976 [6] P Scott The geometries of 3-manifolds Bll London Math Soc, 1983, 15: [7] J H Rbinstein, S C Wang π 1 -injective srfaces in graph manifolds Comm Math Helv, 1998, 73: 1 17 [8] M Hall, Jr Combinatorial Theory Blaisdell Pblishing Company, 1967