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Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction

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Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction Graph manifolds and taut foliations 1 2 Mark Brittenham , Ramin Naimi, and Rachel Roberts UniversityofTexas at Austin Abstract. We examine the existence of foliations without Reeb comp onents, taut 1 1 foliations, and foliations with no S S -leaves, among graph manifolds. We show that each condition is strictly stronger than its predecessors, in the strongest p os- sible sense; there are manifolds admitting foliations of eachtyp e which do not admit foliations of the succeeding typ es. x0 Introduction Taut foliations have b een increasingly useful in understanding the top ology of 3-manifolds, thanks largely to the work of David Gabai [Ga1]. Many 3-manifolds admit taut foliations [Ro1],[De],[Na1], although some do not [Br1],[Cl]. To date, however, there are no adequate necessary or sucient conditions for a manifold to admit a taut foliation. This pap er seeks to add to this confusion. In this pap er we study the existence of taut foliations and various re nements, among graph manifolds. What we show is that there are many graph manifolds which admit foliations that are as re ned as wecho ose, but which do not admit foliations admitting any further re nements. For example, we nd manifolds which admit foliations without Reeb comp onents, but no taut foliations. We also nd 0 manifolds admitting C foliations with no compact leaves, but which do not admit 2 any C such foliations. These results p oint to the subtle nature b ehind b oth top ological and analytical assumptions when dealing with foliations. A principal motivation for this work came from a particularly interesting exam- ple; the manifold M obtained by 37/2 Dehn surgery on the 2,3,7 pretzel knot K . This manifold is a graph manifold, obtained by gluing two trefoil knot exteriors together along their b oundary tori. We show that every essential laminationin M contains a torus leaf, and therefore every essential laminationintersects the image of K in M . This tells us a great deal ab out essential laminations in the exterior of K . This is discussed in Section 5 b elow. The pap er is organised as follows. In Section 1 we give the necessary background on Seifert- b ered spaces and graph manifolds, and intro duce the appropriate nu- merical co ordinates for describing them. In Section 2 we gather the relevant results Key words and phrases. essential lamination, taut foliation, Seifert- b ered space, graph man- ifold, Anosov ow. 1 Research supp orted in part by NSF grant DMS9400651 2 Research supp orted in part by an NSF Postdo ctoral Fellowship Typ eset by A S-T X M E 1 2 M. Brittenham, R. Naimi, and R. Rob erts on foliations and essential laminations to carry out our pro ofs. Section 3 gives the main reults of the pap er, and Section 4 provides the pro ofs. Section 5 discusses surgery on the 2,3,7 pretzel knot. Section 6 nishes with some sp eculations. This researchwas conducted while the authors were visiting the Universityof Texas at Austin in 1994-95. The authors would like to express their appreciation to the faculty and sta at that institution for their hospitality. x1 Coordinates for graph manifolds 1 A Seifert- b ered space M is an S -bundle whose base is a 2-orbifold. More pre- cisely, a Seifert- b ered space b egins with an honest circle bundle M over a compact 0 surface; for our purp oses it will suce to think ab out a compact, orientable, surface, 1 p ossibly with b oundary, crossed with S .To some of the b oundary comp onents of M we then glue a collection of solid tori, so that the meridional direction of each 0 1 solid torus do es not corresp ond to the S -direction on the b oundary of M . The 0 induced foliation of the b oundaries of each of these solid tori by circles extends, in an essentially unique way, to a foliation by circles of the solid torus, so that the core of the solid torus is a leaf. This gives a foliation of M by circles, whose space of leaves - the quotient space obtained by crushing each circle leaf to a p oint - is a 2-orbifold. Its underlying top ological space is called the base surface of the Seifert- b ering of M . The cone p oints of the orbifold corresp ond to the cores of the solid tori; these cores are called the multiple bers of the Seifert- b ering of M . A manifold M is a graph manifold if it there is a collection T of disjointemb edded tori so that the manifold M jT obtained by splitting M op en along T is a not necessarily connected Seifert- b ered space. We assume that the collection T is minimal, in the sense that for no torus T in T is M jTnT a Seifert- b ered space. We adopt the convention that a Seifert- b ered space is not a graph manifold, so T6= ;. Since the b ering of a Seifert- b ered space is essentially unique [S], we can give a more constructive approach to minimality. Thinking in reverse, a graph manifold is obtained by gluing Se ert- b ered spaces together along some of their b oundary tori; the glued tori b ecome the collection of splitting tori T . The collection T is minimal if, in gluing, the homotopy class of the circle b er in one b oundary torus is not identi ed with the class of the b er in the other b oundary torus. The only exceptions to this rule o ccur when some comp onents are solid tori or T I ; for solid tori, minimality requires that the meridion in the b oundary of the solid torus 1 b e glued to the S - b er, and a T I can either b e absorb ed into a comp onentof M if its ends are not glued together, or must have its ends glued together bya 0 map having on the level of H T ; R no integer-valued eigenvectors. 1 Our results will b e stated in terms of the Seifert- b ered pieces making up the graph manifold, and the gluing maps b etween their b oundary tori. To do so, we will need a prop er set of co ordinates. In [S] Seifert develop ed numerical invariants of what he called ` b ered spaces', and gave a complete classi cation of them in terms of these invariants. They describ e the top ological typ e of the base orbifold, and the way that the the regular b ers spin around the multiple b ers. More explicitly, an orientable Seifert- b ered space M can b e describ ed as follows: start with a compact surface F of genus g and Graph manifolds and taut foliations 3 b b oundary comp onents the underlying top ological space of the base orbifold, and drill out k disks one for eachmultiple b er of the Seifert- b ering. To b e sure the resulting surface has non-empty b oundary, drill out one more `zero-th' disk, giving a 1 surface F .Now construct the unique S -bundle M over F with orientable total 0 0 0 space. This bundle has a not necessarily unique cross-section s:F !M b ecause 0 0 @M 6= ;. The images of @F in @M , together with the circle b ers in @M , give 0 0 0 0 us a system of co ordinates for curves in @M , de ning for each simple closed curve 0 in a comp onentof@M a slop e in Q[f1g, where the section de nes slop e 0 and 0 the b er de nes slop e 1.We then glue k + 1 solid tori backonto M to obtain 0 M . The gluing of the i-th solid torus identi es the b oundary of a meridion disk to some curve a b er + b section in @M . These gluings completely describ e the i i 0 Seifert- b ered space, giving us its so-called Seifert invariant M = g,b; a /b ,a /b ,:::,a /b . 0 0 1 1 k k equals + if F is orientable, if not. The rational numb ers a /b are treated i i as an unordered k +1-tuple. The denomenator of each rational numb er in lowest terms turns out to b e the multiplicity of the corresp onding multiple b er. Since our zero-th disk did not corresp ond to a multiple b er, its multiplicityis1,sob =1. 0 This invariant is dep endent up on the choice of section for M ; the only way 0 this section can change, however, is by summing along vertical annuli and tori see Figure 1. Summing along a torus do es not change the asso ciated invariant, and summing along an annulus changes the invariantinavery controlled way; it adds and subtracts 1 each from the invariants asso ciated to the two comp onents of @M 0 containing the b oundary of the annulus. Figure 1 We can actually remove this ambiguityby exploiting it; by a series of summings along annuli one of whose b oundaries lies over the b oundary of the zero-th disk, we can arrange that 0a /b <1, for every i=1,:::,k ; essentially, this amounts to i i gathering the integer parts of the a /b into a /b = a . This gives us a normalized i i 0 0 0 Seifert invariant M = g,b; a /1,a /b ,:::,a /b 0 1 1 k k where a ,b 2Z, and 0<a <b , for i=1,:::,k .
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