SUTURED MANIFOLD HIERARCHIES, ESSENTIAL

LAMINATIONS, AND DEHN SURGERY

Ying-Qing Wu

Abstract. We use sutured manifold theory, essential laminations and essential branched surfaces to establish the upper bounds of distances between certain types of nonsimple Dehn surgery slopes. This is the revised version of an earlier preprint Dehn surgery and simple manifolds.

§0. Introduction

A compact orientable surface F with nonnegative Euler characteristic is either a sphere, a disk, a torus, or an annulus. If a 3-manifold M contains such an essential surface, then it is said to be reducible, ∂-reducible, toroidal, or annular, respectively. Any such surface can be used to decompose the manifold further into simpler manifolds. We say that M is a simple manifold if it has no such surfaces. A simple manifold is expected to have a nice geometric structure. If M has nonempty boundary, then the Geometrization Theorem of Thurston for Haken manifolds says that M with boundary tori removed admits a finite volume hyperbolic structure with totally geodesic boundary. When M has no boundaries, Thurston’s Geometrization Conjecture asserts that M is either hyperbolic, or is a Seifert fiber space with orbifold a sphere with at most 3 cone points.

Suppose T is a torus boundary component of M. We use M(γ) to denote the manifold obtained by Dehn filling on T so that the slope γ on T bounds a disk in the Dehn filling solid torus. When M = E(K) is the exterior of a knot K in S^{3}, denote M(γ) by K(γ), and call it the manifold obtained by γ surgery on the knot K. It is well known that if M is simple then there are only finitely many Dehn fillings on T which produce non simple manifolds. If M(γ_{1}) and M(γ_{2}) are non simple manifolds, then the geometric intersection number between γ_{1} and γ_{2}, denoted by Δ(γ_{1}, γ_{2}), is proved to be at most 8 by Gordon [Gor1]. There are 10 different cases, according to the types of nonnegative Euler characteristic surfaces in M(γ_{i}). In most cases, the upper bounds for Δ(γ_{1}, γ_{2}) have now been established, see Table 0.1.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 57N10, 57M55.

Research at MSRI supported in part by NSF grant #DMS 9022140.

Table 0.1: Upper bounds of Δ(γ_{1}, γ_{2})

In the table, the left column and the top row denote the types of M(γ_{1}) and M(γ_{2}). D, S, A, T mean that M(γ_{i}) contains an essential disk, sphere, annulus, or torus, respectively. All the numbers except the entry of D–A are known to be the best possible, while the bound 3 for D–A is the best result so far (till March 1997), with 2 as its conjectured best possible value. The results can be found in the following papers: The upper bounds for T –T, T –A and A–A are proved by Gordon [Gor1]; the realization of T –A and A–A are very recently noticed by Gordon and Wu [GW]; D–A by Qiu [Q]; S–T is proved independently by Wu [Wu3] and Oh [Oh]; D–D by Wu [Wu2]; S–D by Scharlemann [Sch2]; S–S and D–T by Gordon and Luecke [GLu, GLu3]; S–A is to be proved in this paper:

Theorem 5.1. Suppose M is a simple manifold with torus T as a boundary component. If M(γ_{1}) is annular, and M(γ_{2}) is reducible, then Δ(γ_{1}, γ_{2}) ≤ 2.

The theorem is sharp. Hayashi and Motegi [HM] gave an example of a hyperbolic 3-manifold M, such that M(γ_{1}) is reducible and ∂-reducible, M(γ_{2}) is toroidal and annular, and Δ(γ_{1}, γ_{2}) = 2.

There are many examples showing that a generic hyperbolic manifold admits very few nonhyperbolic Dehn fillings. The following theorem shows that if the manifold is “large”, then stronger results than those in Table 0.1 hold.

Theorems 4.1 and 4.6. Let M be a simple 3-manifold with torus T as a boundary component, such that H_{2}(M, ∂M − T) ≠ 0. If γ_{1} and γ_{2} are slopes on T such that

(1) M(γ_{1}) is annular and M(γ_{2}) is reducible, or

(2) M(γ_{1}) is toroidal, and M(γ_{2}) is reducible, or

(3) M(γ_{1}) is toroidal, and M(γ_{2}) is ∂-reducible, then Δ(γ_{1}, γ_{2}) ≤ 1.

Note that the condition H_{2}(M, ∂M − T) ≠ 0 is true unless M is either a rational homology solid torus or a rational homology cobordism between two tori. In particular, it is true if M either has a boundary component with genus ≥ 2, or if it has more than two boundary tori. Similar to Table 0.1, we have the following table of upper bounds of Δ(γ_{1}, γ_{2}) for such manifolds.

Table 0.2: Upper bounds of Δ(γ_{1}, γ_{2}) when H_{2}(M, ∂M − T) ≠ 0

The entries with question marks are unsettled. The case S–S is proved by Luecke [Lu], others follow from Theorems 4.1, 4.6 and Table 0.1. The results here are also sharp, see Examples 4.7 and 4.8.

The following is an application of the above theorems to Dehn surgery on knots K in S^{3}. It was conjectured that if K is hyperbolic and K(γ) is nonhyperbolic then r = p/q with |q| ≤ 2. Corollary 4.5 proves this conjecture in the case that the knot complement contains an incompressible surface F which is not coannular to itself or ∂N(K).

Corollary 4.5. Let K be a hyperbolic knot in S^{3}. Suppose there is an incompressible surface F in E(K), cutting E(K) into anannular manifolds X and Y. Then K(γ) is hyperbolic for all non-integral slopes γ.

The proofs of these results use a combination of sutured manifold theory, essential laminations, essential branched surfaces, and some combinatorial arguments. In Section 1 we defined cusped manifolds, and show that if there are some essential annuli connecting T to some components of ∂_{h}M, then M(γ) has some nice properties whenever γ has high intersection number with the boundary slope of those annuli (Theorems 1.6, 1.8 and 1.9). Section 2 is devoted to the study of intersections between essential surfaces and essential branched surfaces. We show that they can be modified to intersect essentially on both of them.

Theorem 2.2. Suppose B is an essential branched surface which fully carries a lamination λ, and suppose F is an essential surface in M. Then there is an essential branched surface B′ which is a λ-splitting of B, and a surface F′ isotopic to F, such that F′ ∩ B′ is an essential train track on F′.

This theorem is fundamental in our proofs, and should be useful in the future. Combining with results of Gabai-Mosher and Brittenham it gives the following theorem. Recall that a closed orientable 3-manifold is hyperbolike if it is atoroidal, irreducible, and is not a small Seifert fiber space [Gor2].

Theorem 2.5. If M is a hyperbolic manifold with torus boundary T, then there is a slope γ_{0} on T, called a degenerate slope, such that if Δ(γ_{0}, γ) > 2, then M(γ) is hyperbolike.

By Brittenham [Br2], a combination of this theorem and the 2π-theorem of Gromov-Thurston [GH] gives the result that there are at most 20 non hyperbolike Dehn fillings on a hyperbolic manifold, which improves the bound 24 given by the 2π-theorem [BH]. See Corollary 2.6.

In Section 3 we use sutured manifold hierarchy to prove Theorems 4.1 and 4.6 in the case that ∂M consists of tori. The general version of these theorems is proved in Section 4, using a construction of Luecke. In Section 5 we prove Theorem 5.1. Here we use a combinatorial argument to deal with the cases in which the essential annulus in M(γ_{1}) intersects the Dehn filling solid torus at most twice, then we use β-taut sutured manifold hierarchy and a generalized version of Gabai disk argument to prove the theorem in the general case.

Many problems arise in this study. A challenging problem is to establish the lower limit of Δ(γ_{1}, γ_{2}) for the remaining cases in Tables 0.1 and 0.2. Some other problems can be found in the paper.

Notations and Convensions. All 3-manifolds and surfaces below are assumed orientable. Surfaces in 3-manifoldsare assumed properly embedded unless otherwise stated. 3-manifolds are compact unless it is the exterior of some essential lamination in a compact manifold. If F is a non sphere surface in a 3-manifold M, then it is essential if it is incompressible, ∂-incompressible, and is not parallel to a surface on ∂M. A sphere in M is essential if it is a reducing sphere. We refer the reader to the books of Hempel [He] and Jaco [Ja] for standard definitions and basic results on 3-manifold topology, to the paper of Gabai-Oertel [GO] for those of essential lamination and essential branched surface, and to the papers of Gabai [Ga1–3] and Scharlemann [Sch1–2] for sutured manifold theory.

We use N(B) to denote a regular neighborhood of a set B in M. If F is a surface in M, we use M|F to denote the manifold obtained by cutting M along F, i.e. M|F = M − IntN(F). A train track τ is a branched compact 1-manifold on a surface. A small neighborhood of a branched point is cut into 3 pieces, one of which contains a cusp at that branched point. A branched surface is a 2-dimensional generalization of train tracks, see [GO] for more details. If B is a branched surface in a 3-manifold M, then its exterior is defines as E(B) = M − IntN(B).

Acknowledgement. I would like to thank Martin Scharlemann for his very interesting and stimulating lectures on sutured manifold theory given several years ago at Santa Barbara, which has been of great influence to this work, and thank David Gabai for some helpful conversations on sutured manifolds and essential laminations.

§1. Cusped manifolds

Definition 1.1. A cusped manifold is a compact orientable 3-manifold M with a specified (possibly empty) collection of annuli and tori on ∂M, denoted by ∂_{v}M, called cusps or vertical surfaces. The surface ∂_{h}M = ∂M − Int∂_{v}M is called the horizontal surface of M.

Example 1.2. If B is a branched surface in IntM, then E(B) has a natural cusped manifold structure. The horizontal surface of E(B) is ∂_{h}E(B) = ∂_{h}N(B)∪∂M, and the vertical surfaces of E(B) is ∂_{v}E(B) = ∂_{v}N(B), where ∂_{h}N(B) and ∂_{v}N(B) are the horizontal and vertical boundaries of N(B), see [GO] for definitions. Sutured manifolds are also cusped manifolds, see Section 2.

Definition 1.3. Let F be a surface in M such that F ∩ ∂_{v}M is a set of essential arcs and circles in ∂_{v}M. We consider the arcs of F ∩ ∂_{v}M as cusps on ∂F. The cusped Euler characteristic of F is defined as χ_{c}(F) = χ(F)− \frac{1}{2}C(F), where χ(F) is the Euler characteristic of F, and C(F) is the number of cusps on ∂F. This number has an additivity property: If τ is a train track on F, then χ(F) = ∑ χ_{c}(F_{i}), where F_{i} are the components of F − IntN(τ) . In particular, the formular holds if B is a branched surface in IntM, F a surface transverse to B, and F_{i} the components of F ∩ E(B).

Suppose M is a cusped manifold. A monogon in M is a disk D properly embedded in M, such that ∂D ∩ ∂_{v}M is a single essential arc in ∂_{v}M. In this case D is also called a monogon of the horizontal surface ∂_{h}M. If D ∩ ∂_{v}M consists of two essential arcs in ∂_{v}M, then D is called a bigon. An annulus A properly embedded in M is cuspless if ∂A is disjoint from ∂_{v}M.

Definition 1.4. A cusped manifold M is χ_{c}-irreducible if it contains no essential surface F with χ_{c}(F) > 0. This is equivalent to that (i) M is irreducible, (ii) ∂_{h}M is incompressible, and (iii) M has no monogons.

We say that a branched surface B is intrinsically essential if (i) it has no disk of contact, (ii) it has no Reeb branched subsurface, (iii) no component of ∂_{h}N(B) is a sphere, and (iv) it fully carries a lamination. A branched surface B embedded in M is essential if it is intrinsically essential, and E(B) is χ_{c}-irreducible. The intrinsic part is independent of the embedding of B. Thus to show an essential branched surface B in M remains essential after Dehn filling on a torus T disjoint from B, it would suffice to show that X(γ) is a χ_{c}-irreducible cusped manifold, where X is the component of E(B) containing T.

An essential annulus A in M is called an accidental annulus if it has one boundary component in each of T and ∂_{h}M. The curve A ∩ T is the slope of A on T. Two such annuli A_{1}, A_{2} are parallel if they cut off a product region A_{1} × I containing no cusps of M. We need the following lemma.

Lemma 1.5. If A_{1}, …, A_{n} are mutually nonparallel, mutually disjoint, accidental annuli in a χ_{c}-irreducible cusped manifold M which is not a T^{2} × I with no cusps, then the frontier of X = N(T ∪ A_{1} ∪ … ∪ A_{n}) are essential annuli in M − ∂_{v}M.

Proof. Denote by A′ = A′_{1} ∪ … ∪ A′_{m} the frontier of X. Since A_{i} are essential, A′_{j} are incompressible. Clearly there is no ∂-compressing disk of A′_{j} in X. If there is a ∂-compressing disk D of A′_{j} in X′ = M −IntX disjoint from ∂_{v}M, then the frontier of N(D ∪ A′_{j}) contains a disk Δ. Since M is χ_{c}-irreducible, Δ must be parallel to a disk on ∂_{h}M, which implies that A′_{j} is parallel to an annulus on ∂_{h}M. If n ≥ 2, then the two annuli among the A_{i} which are adjacent to A′_{j} would be parallel. If n = 1, then M is a T^{2} × I without cusp. Both cases contradict the assumptions of the lemma. □

Theorem 1.6. Let M be a χ_{c}-irreducible cusped manifold, and T a torus component of ∂_{h}M. Suppose M is not a T^{2} × I without cusp, and suppose there is an accidental annulus A with slope l on T.

(1) If Δ(l, γ) > 1, then M(γ) is χ_{c}-irreducible;

(2) If Δ(l, γ) > 2, then M(γ) is not an I-bundle over a surface S with ∂_{v}M(γ) the I-bundle over ∂S;

(3) If Δ(l, γ) > 2, any collection of bigons, tori and cuspless annuli in M(γ) can be rel ∂ isotoped into M.

Proof. (1) Put X = N(T ∪ A), and let A′ be the frontier of X. By Lemma 1.5, A′ is essential in M. After Dehn filling, X(γ) is a solid torus with A′ running Δ(l, γ) times along the longitude. So if Δ(l, γ) > 1 then A′ remains an essential annulus in M(γ). By an innermost circle outermost arc argument one can show that M(γ) is χ_{c}-irreducible, hence (1) holds.

(2) Suppose Δ(l, γ) > 2, and suppose M(γ) is an I-bundle over a surface S such that ∂_{v}M(γ) is the I-bundle over ∂S. Since the annulus A′ above is incompressible, with ∂A′ ⊂ ∂_{h}M(γ), it is isotopic to a vertical annulus, so after cutting along A′, the manifold is still an I-bundle over some surface S′, with the two copies of A′ and ∂_{h}M(γ) as the I-bundle over ∂S′. But this is impossible because X(γ) is a solid torus with A′ running along the longitude at least 3 times.

(3) Suppose Δ(l, γ) > 2, and suppose F is a collection of bigons and cuspless annuli in M(γ). Since A′ is essential, by an isotopy we may assume that A′ ∩ F consists of essential arcs and circles in F. Here an arc in a bigon B is essential if each component of B − A′ intersects a cusp of M. Thus each component C of F ∩X(γ) is either an annulus with boundary disjoint from ∂A′ or a disk intersecting A′ twice. If C is an annulus, it can be pushed off the Dehn filling solid torus to lie in M. Since Δ(l, γ) > 2, a meridian disk of X(γ) intersects A′ at least three times, so if C is a disk, it can not be an essential disk of X(γ), hence again it can be isotoped into M.

By restricting the above isotopies to M − N(∂M) and extending continuously over M, we may assume that the restriction of the above isotopies to ∂M are identity isotopies. □

Question 1.7. Assuming all conditions of Theorem 1.6 except the existence of the annulus A, is there still a slope l on T such that the conclusions of the theorem hold?

If in the above theorem we can find two different annuli in M, then a stronger result holds:

Theorem 1.8. Let M be a χ_{c}-irreducible cusped manifold, and T a torus component of ∂_{h}M. Suppose there are two disjoint, nonparallel, accidental annuli A_{1} and A_{2} in M with slope l on T.

(1) If γ ≠ l, then M(γ) is χ_{c}-irreducible;

(2) If Δ(l, γ) > 1, then M(γ) is not an I-bundle over a surface S with ∂_{v}M(γ) the I-bundle over ∂S;

(3) If Δ(l, γ) > 1, any collection of bigons, tori and cuspless annuli in M(γ) can be rel ∂ isotoped into M.

Proof. The proof is similar to that of Theorem 1.6, only that we take X = N(T ∪ A_{1} ∪ A_{2}). The frontier of X now consists of two annuli A′ and A′′, which are essential in M by Lemma 1.5. If γ ≠ l then A′ ∪ A′′ is essential in X(γ), and if Δ(l, γ) > 1 then a meridian disk of X(γ) intersects A′ ∪ A′′ more than two times, hence the argument in the proof of Theorem 1.6 applies here. □

The “two annuli” condition was first applied by Menasco to show that essential surfaces remain essential after Dehn surgery on certain knots [Me]. If there are three such annuli in M, then we can get the strongest possible conclusion. The proof of the following theorem is similar to that of Theorem 1.8, and is omitted.

Theorem 1.9. If in Theorem 1.8 there are three disjoint, nonparallel, accidental annuli A_{i}, then the conclusions of (1), (2) and (3) in Theorem 1.8 hold for all γ ≠ l. □

§2. Intersection between essential surface and essential lamination

Given two essential surfaces F_{1}, F_{2}, it is always possible to isotope one of them so that they intersect in circles essential on both surfaces. This is not possible in general if one of them is an essential branched surface B. However, Theorem 2.2 shows that after some splitting of B, it is possible to make them intersect essentially. As a corollary, it is shown that we can isotope an essential surface F so that its intersection with an essential lamination in M^{3} is an essential lamination on F. The results will be applied in Section 3 to prove Theorems 3.3 and 3.4. As a byproduct, we will combine our result with those of Gabai-Mosher and Brittenham to prove that all but 5 lines of surgeries on a hyperbolic knot produce hyperbolike manifolds.

Lemma 2.1. Suppose λ is an essential lamination fully carried by a branched surface B. Then λ is fully carried by an essential branched surface B′ which is a λ-splitting of B.

Proof. This follows from the proofs of Lemma 4.3 and Proposition 4.5 in [GO]. The argument goes as follows. By thickening λ if necessary we may assume that ∂_{h}N(B) ⊂ λ. By a λ-splitting we may assume that B has no compact surface of contact. Since λ is essential, E(B) is irreducible, and ∂_{h}E(B) is incompressible in E(B). Also, a monogon of E(B) could be extended to a monogon for λ via some half-infinite vertical strip in N(B)−λ, which would contradict the essentiality of λ. It follows that the branched surface B satisfies all conditions of an essential branched surface except possibly the condition that it has no Reeb branched surfaces. Since λ is essential, it has no vanishing cycle, so [GO, Lemma 4.3] says that λ is also fully carried by an essential branched surface B′. By examining the proof of that lemma, one can see that B′ is actually obtained by a λ-splitting of B. □

Theorem 2.2. Suppose B is an essential branched surface which fully carries a lamination λ, and suppose F is an essential surface in M. Then there is an essential branched surface B′ which is a λ-splitting of B, and a surface F′ isotopic to F, such that F′ ∩ B′ is an essential train track on F′.

Proof. By an isotopy we may assume that F is transverse to B. Thus F ∩ B is a train track τ on F. We may further assume that F ∩ N(B) = N(τ), and the I-fibers of N(τ) are also I-fibers of N(B), see [GO, Lemma 2.6]. The train track τ can not have any monogon, because a monogon bounded by τ would also be a monogon for the exterior of B, which is impossible since B is essential. We need to modify B and F so that τ has no 0-gons either.

Suppose D is a 0-gon of τ in F. Let D_{1} be a small neighborhood of D in F. Then the train track τ in D_{1} consists of a circle C bounding the 0-gon D, and some arcs from some branch points on C to the boundary of D_{1}, so it looks like that in Figure 2.1(a). The foliation F of N(B) induces a foliation F_{1} = F ∩F on N(τ), see Figure 2.2(b) for an example. We claim that F_{1} has no noncompact leaf. Otherwise, there would be a circle γ which is the limit of noncompact leaves. Let l be the leaf of F that contains γ. Since the lamination λ is essential, l is π_{1}-injective, so γ is a trivial loop on l, which contradicts the Reeb stability of λ, (see [GO, Lemma 2.2].)

Figure 2.1

It follows that there is an annulus γ × I in D_{1}, such that each γ × t is a leaf of F , γ × 0 = ∂D, and γ × 1 intersects ∂_{h}N(τ). Since λ is essential, γ × 1 bounds a disk D′ in a leaf l which intersects ∂_{h}N(B), so there is an interstitial I-bundle J in M which contains the part of D′ that is not on ∂_{h}N(B). By splitting along a compact subbundle of J that contains J ∩ D′, we get a new branched surface B′, such that τ′ = B′ ∩ F is the train track obtained by splitting τ along the γ × 1, as shown in Figure 2.1(c). Since M is irreducible (because it contains an essential lamination), D ∪ D′ bounds a 3-ball, so F is isotopic to a surface F′ such that B′ ∩ F′ = B ∩ F′ − γ. By induction on the number of components in F − B, we can eliminate all 0-gons, so eventually F′ ∩ B′ is an essential train track on F′, as requires.

By Lemma 2.1, B′ can be further λ-split to an essential branched surface B′′. The train track B′′ ∩ F′ is a splitting of B′ ∩ F′, so it is still essential. □

Corollary 2.3. If λ is an essential lamination, and F is an essential surface in M, then F can be isotoped so that λ∩F is an essential lamination on F. In particular, for any leaf l in λ, each circle component of l ∩ F is essential on both F and l.

Proof. By the theorem there is an essential branched surface B′ fully carrying λ, such that B′ ∩ F is an essential train track on F. Since λ ∩ F is fully carried by the essential train track B ∩ F, it is an essential lamination on F. If C is a circle component of λ ∩ F, then it is essential on F. Since F is incompressible, C must also be essential on the leaf of λ that contains C. □

Question 2.4. Can the corollary be generalized to the intersection of two essential laminations?

Following Gordon [Gor2], we say that a closed manifold M is hyperbolike if it is irreducible, atoroidal, and is not a small Seifert fiber space. Thurston’s geometrization conjecture [Th] asserts that hyperbolike manifolds are hyperbolic. The following theorem says that except for five lines in the Dehn surgery space, all surgeries on a hyperbolic knot are hyperbolike. Its proof uses results of Gabai-Mosher on the existence of essential laminations on knot complements, Brittenham’s criterion of small Seifert fiber spaces, and Theorem 2.2.

Theorem 2.5. If M is a hyperbolic manifold with boundary a torus T, then there is a slope γ_{0} on T, called a degenerate slope, such that if Δ(γ_{0}, γ) > 2, then M(γ) is hyperbolike.

Proof. Gabai and Mosher [GM] showed that there is an essential branched surface B in M, such that each component of E(B) = M − IntN(B) is either a solid torus or a manifold X = T × I with nonempty set of cusps on T × 1. The cusps on T × 1 are isotopic to a slope γ_{0} on T = T × 0. It is clear that B remains essential in M(γ) whenever Δ(γ_{0}, γ) ≥ 2; in particular, M(γ) is irreducible. Brittenham [Br2] observed that if Δ(γ_{0}, γ) > 2, then X(γ) is not an I-bundle over a surface F with cusps the I-bundle over ∂F, so by the result of [Br1], such manifold is not a small Seifert fiber space. To prove the theorem, it remains to show that M(γ) is atoroidal when Δ(γ_{0}, γ) ≥ 3.

Assuming the contrary, let F be a torus in M(γ). Let λ be an essential lamination fully carried by B. By Theorem 2.2, there is an essential branched surface B′ which λ-splits B, and a surface isotopic to F (still denoted by F), such that B′ ∩ F is an essential train track on F. Let X′ be the component of M − IntN(B′) containing T. Notice that X′ can not be a T × I without cusp, otherwise T × 1 would be a leaf of λ, so λ would be inessential after all Dehn fillings on M; but since λ is fully carried by B, which is essential in M(γ) when Δ(γ_{0}, γ) > 2, this is impossible.

There is an annulus A in X with one boundary component ∂_{0} on T with slope γ_{0}, and the other boundary component ∂_{1} on ∂_{h}X −T. Since B′ is a splitting of B, the annulus A also lives in X′, with ∂_{1} on ∂_{h}X′ − T. Since B′ ∩ F is an essential train track on F, each component of F ∩ X′(γ) is either a bigon or a cuspless annulus in X′(γ). According to Theorem 1.6, the surface F ∩ X′(γ) can be rel ∂ isotoped into X′, which implies that F is isotopic to a torus in M. This contradicts the assumption that M is atoroidal, completing the proof. □

Corollary 2.6. Let M be a hyperbolic manifold with boundary a torus T. Then M(γ) is hyperbolike for all but at most 20 slopes γ.

Proof. This follows from Theorem 2.5 and the proof of [Br2]. Brittenham showed that there are at most 20 slopes γ which has Δ(γ, l) ≤ 2 and has length at most 2π on T, and they contain all the reducible or small Seifert fibered slopes. Theorem 2.5 says that this set also contains all the toroidal slopes. □

§3. Sutured manifold decomposition and essential branched surfaces

A sutured manifold is a triple (M, γ, β), where M is a compact orientable 3manifold, γ a set of annuli or tori on ∂M, and β a proper 1-complex in M. The pair (M, γ) is a cusped manifold M such that each component of ∂_{h}M is oriented + or −, and each annulus cusp is adjacent to two components of ∂_{h}M with different orientation. In this case the cusps γ are called sutures in [Ga1, Sch1]. Denote by ∂_{+}M (resp. ∂_{−}M) the union of all components of ∂_{h}M with + (resp. −) orientation. We use ∂_{±}M to denote “∂_{+}M or ∂_{−}M”. They are denoted by R_{±} in [Ga1, Sch1]. In this section we assume β = ∅.

A sutured manifold M is taut if it is χ_{c}-irreducible, and both ∂_{±}M minimize the Thurston norm in H_{2}(M, ∂_{v}M). Note that since each annular cusp is adjacent to two different component of ∂_{h}M, it is automatically true that M has no monogon.

If F is an oriented surface properly embedded in M, such that ∂F intersects each torus component of ∂_{v}M in coherently oriented circles, then when cutting along F, the manifold M_{1} = M|F = M − IntN(F) has a natural sutured manifold structure (M_{1}, γ_{1}), see [Ga1, Sch1]. Such process of obtaining a new sutured manifold from the old one by cutting along an oriented surface is called a sutured manifold decomposition, and is denoted by (M, γ) ⇝^{F} (M_{1}, γ_{1}). The decomposition is taut if both M and M_{1} are taut sutured manifolds.

Theorem 3.1 (Gabai). Let M be a Haken 3-manifold with toroidal boundary. Let P be a specified component of ∂M. Suppose M is atoroidal, and H_{2}(M, ∂M −P) ≠ 0. Then there exists a sequence

(M, P) = (M_{0}, γ_{0}) ⇝^{S_{1}} (M_{1}, γ_{1}) ⇝^{S_{2}} ⋯ ⇝^{S_{n}} (M_{n}, γ_{n})

SUTURED MANIFOLDS, ESSENTIAL LAMINATIONS, AND DEHN SURGERY 11

of sutured manifold decompositions with the following properties:

(1) Each (M_{i}, γ_{i}) is taut and each separating component of S_{i+1} is a product disk;

(2) Some component of γ_{n} is the torus P;

(3) (M_{n}, γ_{n}) is a union of a product sutured manifold and a sutured manifold (H, δ) = T^{2} × I where P = T^{2} × 0, and δ ∩ (T^{2} × 1) ≠ ∅.

Proof. When ∂M = P, this is exactly Step 1 in the proof of [Ga2, Theorem 1.7]. Since ∂M is incompressible, the sutured manifold (M, P) is taut, with ∂_{+}M = ∂M −P. Hence (M, P) satisfies the assumption of [Ga2, Theorem 1.8]. As remarked in the paragraph before [Ga2, Theorem 1.8], the proof of Theorem 1.7 there applies verbatim to this more general setting. □

Given any sequence of sutured manifold decomposition of M, there is an associated branched surface B. The construction is obvious: B is the union of ∂M − P and all the S_{i} in the sequence, smoothed at ∂S_{i} according to its orientation. See [Ga3, Construction 4.16] for details. The following theorem is also due to Gabai.

Theorem 3.2. The branched surface B associated to the sequence in Theorem 3.1 fully carries an essential lamination λ.

Proof. The construction of λ was described in [Ga3, Construction 4.17]. The lamination extends to taut foliations after all but one Dehn filling on P, so it is essential in all but one M(γ). It follows that λ is also essential in M.

One can also prove the essentiality of λ directly from the construction. From Description 2 of [Ga3, Construction 4.17], we see that the only compact leaves of λ are ∂M − P, which are incompressible by our assumption. As usual, let M_{λ} = M − Intλ_{1}, where λ_{1} is a thickening of λ. Then M_{λ} is the union of E(B) and a noncompact product sutured manifold (W, β) along the annular sutures β of B. Since E(B) is a taut sutured manifold, E(B) is χ_{c}-irreducible, so M_{λ} is irreducible, and ∂M_{λ} is incompressible and end-incompressible. By definition λ is essential. □

Theorem 3.3. Let M be an atoroidal, irreducible, ∂-irreducible, compact 3-manifold with ∂M a set of tori. Let T be a specified component of ∂M. Suppose H_{2}(M, ∂M − T) ≠ 0. Let γ_{1} and γ_{2} be slopes on T such that M(γ_{1}) is toroidal, and M(γ_{2}) is reducible or ∂-reducible. Then Δ(γ_{1}, γ_{2}) ≤ 1.

Proof. Let

(M, ∂M) = (M_{0}, γ_{0}) ⇝^{S_{1}} (M_{1}, γ_{1}) ⇝^{S_{2}} ⋯ ⇝^{S_{n}} (M_{n}, γ_{n})

be a sequence of sutured manifold decomposition given by Theorem 3.1. Let B and λ be the branched surface and essential lamination given by the Theorem. By Lemma 2.1 B can be λ-split into an essential branched surface B′. Let X (resp. X′) be the component of E(B) (resp. E(B′)) containing T. Note that since B′ is a splitting of B, we have X ⊂ X′.

According to Theorem 3.1(3), X is a sutured manifold T^{2} × I with T = T^{2} × 0, and ∂_{v}X ∩ (T^{2} × 1) ≠ ∅. Thus ∂_{v}X consists of (at least two) annuli, cutting T^{2} × 1 into ∂_{+}X and ∂_{−}X. Hence there are two essential annuli A_{±} in X, each having one boundary component on T, and the other on different components of ∂_{±}X. After splitting the component X′ may no longer be a T^{2} × I, but since X ⊂ X′, the above annuli A_{±} are also essential annuli in X′. Let γ′ be the slope A_{+} ∩ T on T. Applying Theorem 1.8 to X′, we see that if γ ≠ γ′ then X′(γ) is χ_{c}-irreducible, hence B′ remains an essential branched surface in X′(γ) for all γ ≠ γ′. Since we assumed that M(γ_{2}) is reducible or ∂-reducible, we must have γ′ = γ_{2}.

Now consider an essential torus F in M(γ_{1}). By Lemma 2.1, after an isotopy of F we may assume that there is a branched surface B′′ which is a λ-splitting B′, such that F ∩ B′′ is an essential train track τ on F. Since the Euler number of F is zero, all components of F − IntN(τ) are bigons or cuspless annuli. Let X′′ be the component of E(B′′) that contains T. Then in particular Y = F ∩ X′′(γ) is a union of bigons and cuspless annuli. By Theorem 1.8, if Δ(γ_{1}, γ_{2}) ≥ 2 then Y can be rel ∂ isotoped into M. But then F would be isotopic to an essential torus in M, contradicting the assumption that M is atoroidal. □

Theorem 3.4. Let M be a simple compact 3-manifold with ∂M a set of tori. Let T be a specified component of ∂M. Suppose H_{2}(M, ∂M − T) ≠ 0. Let γ_{1} and γ_{2} be slopes on T such that M(γ_{1}) is annular, and M(γ_{2}) is reducible or ∂-reducible. Then Δ(γ_{1}, γ_{2}) ≤ 1.

Proof. The proof is the same as that of Theorem 3.3, with the essential torus F in M(γ_{1}) replaced by an essential annulus A. Since ∂M(γ_{1}) is contained in the branched surface B′′ in that proof, we can assume that A ∩ B′′ is an essential train track τ in A containing ∂A. Hence A − IntN(τ) consists of bigons and cuspless annuli. So if Δ(γ_{1}, γ_{2}) ≥ 2 then A can be rel ∂ isotoped into M, which would contradict the assumption that M is simple. □

§4. Surgery on manifolds with large boundary

In [Wu3] and [Oh] it was proved that if M is a hyperbolic manifold with T a torus boundary component, and if M(γ_{1}) is reducible and M(γ_{2}) toroidal, then Δ(γ_{1}, γ_{2}) ≤ 3. Theorem 3.3 says that if ∂M consists of tori, and H_{2}(M, ∂M −T) ≠ 0, then we actually have a much stronger conclusion that Δ(γ_{1}, γ_{2}) ≤ 1. The following theorem shows that the first condition can be removed. Note that any manifold M whose boundary is not one or two tori automatically satisfies the condition that H_{2}(M, ∂M − T) ≠ 0.

Theorem 4.1. Let M be an irreducible, ∂-irreducible, atoroidal 3-manifold with torus T as a boundary component, such that H_{2}(M, ∂M − T) ≠ 0. Let γ_{1} and γ_{2} be slopes on T such that M(γ_{1}) is toroidal, and M(γ_{2}) is reducible or ∂-reducible. Then Δ(γ_{1}, γ_{2}) ≤ 1.

Proof. First assume that M(γ_{2}) is reducible. Since M is atoroidal, by a theorem of Scharlemann [Sch2], we see that M(γ) remains ∂-irreducible if γ ≠ γ_{2}.

For each component F of ∂M with genus ≥ 2, choose a simple manifold M_{F} with ∂M_{F} = F, and H_{2}(M_{F} , ∂M_{F}) ≠ 0. Such a manifold can be constructed as follows: Let g =genus(F). Choose a compact manifold X such that ∂X is a surface of genus g − 1, and H_{2}(X) has rank ≥ 3. According to Myers [My], there is an arc α in X, such that Y = X − IntN(α) is a simple manifold. Clearly, ∂Y = F. By calculating the homology one can show that H_{2}(Y, ∂Y) ≠ 0. Hence we can take M_{F} = Y.

Gluing an M_{F} to M along F for each nontorus boundary component F of M, we get a manifold M̂, which is atoroidal and Haken, with H_{2}(M, ∂M −T) ≠ 0, and ∂M̂ consisting of tori. If S is a reducing sphere in M(γ_{2}), then since H_{2}(M_{F} , ∂M_{F}) ≠ 0, gluing M_{F} to M(γ_{2}) − IntN(S) will not produce a 3-ball bounded by S. Hence S remains a reducing sphere in M̂(γ_{2}). Since ∂M is incompressible in M(γ_{1}), an essential torus P in M(γ_{1}) remains essential in M̂(γ_{1}), so M(γ_{1}) is toroidal. The result then follows from Theorem 3.3.

Now assume that M(γ_{2}) is ∂-reducible. Let F be a component of ∂M − T which is compressible in M(γ_{2}), and let C be a curve on F bounding a compressing disk D in M(γ_{2}). If F is a torus, then the frontier of N(F ∪ D) is a 2-sphere S. If S is an essential sphere, then M(γ_{2}) is reducible, and the result has been proved above. If S is inessential, then ∂M consists of two tori, so the result follows from Theorem 3.3. Hence we can assume that F has genus ≥ 2.

Let P be a planar surface having at least three boundary components. Let φ : ∂P × I → F be a map such that ∂P is sent to curves parallel to C. Denote by Y = M ∪_{φ} (P × I), the manifold obtained by gluing P × I to M using φ as gluing map. By a standard innermost circle outermost arc argument it can be shown that Y is irreducible, ∂-irreducible and atoroidal. Since ∂Y − T is not a torus, we also have H_{2}(Y, ∂Y − T) ≠ 0. The surface P extends to a sphere in Y (γ_{2}) having some boundary components on each sides, hence Y (γ_{2}) is reducible. If the annuli ∂P × I is incompressible in M(γ_{1}) then an essential torus in M(γ_{1}) remains essential in Y (γ_{1}). Hence the result follows from the first case proved above, with M replaced by Y. If ∂P × I is compressible in M(γ_{1}), then Y (γ_{1}) is also reducible. By the Reducible Surgery Theorem of Gordon-Luecke [GLu], we also have Δ(γ_{1}, γ_{2}) ≤ 1. □

Remark 4.2. The idea of gluing a large simple manifold to M to get the result is due to John Luecke. If M is an irreducible atoroidal manifold with torus boundaries, such that H_{2}(M, ∂M − T) ≠ 0, then a theorem of Gabai [Ga2, Corollary 2.4] says that at most one Dehn filling on T could be reducible. Using the above trick, Luecke showed that this is true even if ∂M has some higher genus components.

Example 4.3. (1) If W is a solid torus, and K is a hyperbolic knot in W with winding number 0, then M = W − IntN(K) satisfies the conditions of Theorem 3.3 and 3.4. Hence if γ surgery on K produces toroidal or annular manifold, then γ is a longitudinal slope, i.e, Δ(γ, m) = 1, where m is the meridional slope of K. Together with results of [Wu2] and [Sch2], it shows that non-integral surgeries on such knots always produce hyperbolic manifolds.

(2) As noticed above, the condition H_{2}(M, ∂M − T) is true unless ∂M − T is empty or a single torus. This condition can not be removed. Hayashi and Motegi [HM] have an example of a simple manifold M, such that ∂M is a union of two tori, M(γ_{1}) is reducible and ∂-reducible, M(γ_{2}) is toroidal and annular, and Δ(γ_{1}, γ_{2}) = 2.

(3) If W is a handlebody of genus ≥ 2, and K is a knot in W such that W − IntN(K) is irreducible, ∂-irreducible and atoroidal, then only integral surgeries on K can yield toroidal manifolds.

Question 4.4. Are there any hyperbolic knot in a solid torus which admits some non-integral toroidal or annular surgeries?

Corollary 4.5. Let K be a hyperbolic knot in S^{3}. Suppose there is an incompressible surface F in E(K), cutting E(K) into anannular manifolds X and Y. Then K(γ) is hyperbolic for all non-integral slopes γ.

Proof. Let X be the component of E(K) − IntN(F) containing T = ∂N(K). Let γ be a non-integral slope on N(K), and let m be the meridional slope. Clearly, F is compressible in X(m). Therefore, by [Wu2] and [Sch2], X(γ) is irreducible and ∂-irreducible. Hence K(γ) = Y ∪ X(γ) is a Haken manifold. Since Y is anannular, any essential torus S in K(γ) can be isotoped to be disjoint from F. Since K is hyperbolic, S can not be in Y, otherwise it would be an essential torus in E(K). By Theorem 4.1, X(γ) is atoroidal, so S can not be in X(γ) either. Thus K(γ) is also atoroidal. If K(γ) were a Seifert fiber space, then F would be isotopic to a surface transverse to the fibers, hence K(γ) − IntN(F) would be an I-bundle over surface, which is impossible because X is anannular. It now follows from Thurston’s Geometrization Theorem [Th] that K(γ) is hyperbolic. □

By a theorem of Gordon [Gor1, Theorem 1.1], if Y (γ_{1}) and Y (γ_{2}) are non simple, then Δ(γ_{1}, γ_{2}) ≤ 5. Since γ_{i} are integral, there are at most 6 such slopes. Hence those knots K in Corollary 4.5 admit at most 6 nontrivial, nonhyperbolic surgeries.

Theorem 4.6. Let M be a compact simple 3-manifold with torus T as a boundary component, such that H_{2}(M, ∂M − T) ≠ 0. Let γ_{1} and γ_{2} be slopes on T such that M(γ_{1}) is annular, and M(γ_{2}) is reducible. Then Δ(γ_{1}, γ_{2}) ≤ 1.

Proof. Let A be an essential annulus in M(γ_{1}). If ∂M consists of tori, the theorem follows from Theorem 3.4. If ∂A lies on torus boundary components of M, we can glue M_{F} to M to get a simple manifold Y with toroidal boundary, as in the proof of Theorem 4.1, then apply Theorem 3.4. Hence we may assume that A has at least one boundary on a nontorus component of ∂M.

Suppose both components of ∂A lie on nontorus boundary components of M. Let X = P × I, where P is an annulus. Gluing (∂P) × I to a neighborhood of ∂A on ∂M, we get a manifold Y. One can check that Y is still a simple manifold, H_{2}(Y, ∂Y − T) ≠ 0, the manifold Y (γ_{1}) is toroidal, and Y (γ_{2}) is reducible. Hence the result follows from Theorem 4.1.

Now suppose that ∂A has one component on a non-torus boundary component F of M, and the other on a torus G on ∂M. Notice that the above construction fails, because then G pushed into M would be an essential torus in Y. We proceed as follows: Let A_{1}, A_{2} be two parallel copies of A. Glue the above manifold X = P ×I to M with ∂P × I identified to a neighborhood of the two curves of ∂A_{i} on the nontorus component F of ∂M. Then the resulting manifold Y again satisfy all the conditions of the theorem, and A_{1} ∪ A_{2} ∪ P is an essential annulus in Y (γ_{1}) with both boundary on the torus G. Hence the result follows from the first case proved above. □

The following examples show that the results of Theorems 4.1 and 4.6 are the best possible.

Example 4.7. Let M be the exterior of the Borromean ring L shown in Figure 4.1, and let T be a specified component of ∂M. It is well known that M is hyperbolic. The trivial surgery M(m) is reducible and ∂-reducible, and the longitudinal surgery M(l) is toroidal because a component of L bounds a punctured torus disjoint from the other components, which extends to a torus F in M(l). Notice that F is non separating, so if it were compressible in M(l), then M(l) would be rducible, which would contract the theorem of Gabai that M admits at most one reducible Dehn fillings [Ga2, Corollary 2.4].

Figure 4.1

Example 4.8. It is more difficult to construct an example of large manifold M with M(γ_{1}) annular and M(γ_{2}) reducible, and Δ(γ_{1}, γ_{2}) = 1. Here is a sketch of such an example.

Let L = K_{1} ∪ K_{2} be the link in a hendlebody H as shown in Figure 4.2(a). Let M_{1} = H − IntN(L), with T_{i} = ∂N(K_{i}).

Figure 4.2

It is easy to show that M_{1} is irreducible, ∂-irreducible, and atoroidal. There is an essential annulus, however, from T_{2} to ∂H. We need to modify the manifold to make it anannular.

Let C_{1}, C_{2} be the curves on ∂H as shown in Figure 4.2(b). Then P = ∂H − IntN(C_{1} ∪ C_{2}) is a sphere with 4 punctures. Choosing a simple manifold X with ∂X a genus 2 surface, and gluing it to M_{1} along P, we get a M. One can show that M is a simple manifold.

Let m be the meridian of K_{1} on T_{1}, and let l be the longitude, i.e the blackboard framing slope. Clearly, M(m) is reducible. We claim that C_{1} ∪ C_{2} bounds an essential annulus in M(l). To see the annulus, choose a Möbius band F_{i} on each handle of H, with ∂F_{i} = C_{i}. Tubing F_{1} and F_{2} together, we get a twice punctured Klein bottle F. Isotope F so that it contains K_{1} and is disjoint from K_{2}. Then F ∩ M is a twice punctured annulus, which can be capped off in M(l) to become an annulus A bounded by C_{1} ∪ C_{2}. Since C_{1} and C_{2} are on different components of ∂M, A is ∂-incompressible. It must also be incompressible, otherwise M(l) would be ∂-reducible; but since M(m) is reducible, this would contradict Scharlemann’s Theorem [Sch].

Question 4.9. (1) If M(γ_{1}) in Theorem 4.6 is ∂-reducible instead of reducible, is the theorem still true? It is true if the boundary of a ∂-reducing disk lies on a torus.

(2) If both M(γ_{i}) are toroidal or annular, what is the upper bound of Δ(γ_{1}, γ_{2})? For general M, Gordon [Gor1] proved that Δ ≤ 8, and Δ ≤ 5 if ∂M ≠ ∅. With the extra assumption that H_{2}(M, ∂M − T) ≠ 0, the upper bound could be much smaller.

§5. Annular surgery and toroidal surgery

Theorem 5.1. Suppose M is a simple manifold with torus T as a boundary component. If M(γ_{1}) is annular, and M(γ_{2}) is reducible, then Δ(γ_{1}, γ_{2}) ≤ 2.

This whole section is devoted to the proof of this theorem. By Theorem 4.6, we may assume that ∂M −T is a torus. We may also assume that M(γ_{1}) is irreducible and ∂-irreducible, otherwise the result follows from the Reducible Surgery Theorem of Gordon and Luecke [GLu] or Scharlemann’s theorem [Sch]. We will further assume Δ(γ_{1}, γ_{2}) ≥ 3. The theorem will follow from the contradiction in the conclusions of Lemma 5.6 and 5.7.

Let F_{1} be an essential annulus in M(γ_{1}). Let F_{2} be either a reducing sphere in M(γ_{2}), or a disk embedded in IntM(γ_{2}). Denote by J_{i} the attached solid torus in M(γ_{i}). Let P_{i} = F_{i} ∩ M. Let u_{1}, …, u_{n_{1}} (resp. v_{1}, …, v_{n_{2}}) be the disks of F_{1} ∩ J_{1} (resp. F_{2} ∩J_{2}), labeled successively when traveling along J_{i}. Let Γ_{1} be the graph in F_{1} with u_{i} as (fat) vertices, and the arc components of P_{1} ∩ P_{2} as edges. Similarly, Γ_{2} is a graph in F_{2} with v_{j} as vertices and the arcs of P_{1} ∩ P_{2} as edges. Notice that if F_{2} is a disk, and e is an arc component of P_{1} ∩ P_{2} with an end on ∂F_{2}, then that end of e is not attached to any fat vertices. We say that e is a ghost edge, so Γ_{i} are actually graphs with ghost edges. The end of e which is not on a vertex is called a ghost end. On F_{2} all ghost ends are on ∂F_{2}, while on F_{1} the ghost ends are in the interior of P_{1}.

Each vertex of Γ_{i} is given a sign according to whether the J_{i} passes F_{i} from the positive side or negative side at this vertex. Two vertices of Γ_{i} are parallel if they have the same sign. Otherwise they are antiparallel. An edge of Γ_{i} is a positive edge if it connects parallel vertices. Otherwise it is a negative edge. The parity rule of [CGLS] says that an edge of P_{1} ∩ P_{2} is a positive edge in Γ_{1} if and only if it is a negative edge in Γ_{2}.

A trivial loop in Γ_{i} is an edge cutting off a disk in P_{i} which contains no vertices of Γ_{i}. Such a disk can be used to ∂-compress the surface F_{j}, j ≠ i. We choose F_{1} so that n_{1} is minimal, which guarantees that Γ_{2} has no trivial loops. In the following, F_{2} is either a reducing sphere of M(γ_{2}), or a disk in the interior of M(γ_{2}) such that all vertices of Γ_{2} are parallel. In the first case, we choose n_{2} to be minimal among all reducing spheres. In the second case, by the parity rule Γ_{1} can not have any loops. In any case, we have that neither Γ_{i} has any trivial loop. Doing some disk swappings if necessary, we may also assume that all circle component of P_{1} ∩P_{2} are essential in both P_{i}. In particular, each disk face of Γ_{i} has interior disjoint from P_{j}, j ≠ i.

We may assume that each circle ∂u_{i} intersects each ∂v_{j} exactly Δ times. If e is an edge of Γ_{1} with an end x on a fat vertex u_{i}, then x is labeled j if x is in u_{i} ∩ v_{j}. The labels in Γ_{1} are considered mod n_{2} integers. In particular, n_{2} + 1 = 1. When going around ∂u_{i}, the labels of the ends of edges appear as 1, 2, …, n_{2} repeated Δ times. Label the ends of edges in Γ_{2} similarly. Each label in Γ_{2} is a mod n_{1} integer. Ghost ends are not labeled.

A set of positive edges {e_{1}, …, e_{k}} on Γ_{i} is called a Scharlemann cycle if (1) they bounds a disk on P_{i} with interior disjoint from Γ_{i}, (2) all the vertices on the ends of e_{j} are parallel, and (3) the two labels at the ends of e_{j} are the same as that of e_{1}. The two labels of e_{i} must be {j, j + 1} for some j. We call {j, j + 1} the label pair of the Scharlemann cycle.

If {e_{1}, e_{2}, e_{3}, e_{4}} are four parallel positive edges with e_{i} adjacent to e_{i+1} for i = 1, 2, 3, and if the two middle edges {e_{2}, e_{3}} form a Scharlemann cycle, then the set of these four edges is called an extended Scharlemann cycle. This is enough for our purpose. We refer the reader to [GLu] for more general definition.

Lemma 5.2. Suppose F_{2} is a reducing sphere. Then (1) Γ_{1} can not have n_{2} parallel edges;

(2) Γ_{1} can not have an extended Scharlemann cycles;

(3) Any two Scharlemann cycles on Γ_{1} have the same label pair;

(4) Γ_{1} can not have more than (n_{2}/2) + 1 parallel positive edges;

(5) If Γ_{1} has a Scharlemann cycle, then F_{2} bounds a punctured lens space. In particular it is separating.

Proof. (1) is proved by Gordon and Litherland in [GLi, Proposition 1.3]. (2)–(4) follow from the proof of [Wu1, Lemma 2.2–2.4].

(5) is well known: If Γ_{1} has a Scharlemann cycle then we can find another sphere F′_{2} which has two fewer intersections with Dehn filling solid torus, and cobounds with F_{2} a punctured lens space. By the minimality of n_{2} the surface F′_{2} must bounds a 3-ball. See the proof of [CGLS, Lemma 2.5.2(a)] for more details. □

Note that since M is a simple manifold, we have n_{2} > 2.

Lemma 5.3. Theorem 5.1 is true if n_{1} ≤ 2.

Proof. If n_{1} = 1, all edges of Γ_{1} are parallel. Since Δ ≥ 3, this contradicts Lemma 5.2(1).

Suppose n_{1} = 2. Consider the reduced graph Γ̂_{1} obtained by replacing a family of parallel edges in Γ_{1} with a single edge. By calculating the Euler number, one can see that Γ̂_{1} has at most 4 edges. Since by Lemma 5.2(1) there is no n_{2} parallel edges, each of the two vertices in Γ̂_{1} must have valency 4, so the graph looks like that in Figure 5.1(a). The edges a and d are positive edges, hence each represents at most (n_{2}/2) + 1 edges in Γ_{1}. Since each of b and c represents at most n_{2} − 1 edges, and since the valency of each vertex in Γ_{1} is 3n_{2}, it follows that each of a and d represents exactly (n_{2}/2) + 1 edges, and each of b and c represents n_{2} − 1 edges. See Figure 5.1(b) for the case that n_{2} = 6. We separate two cases.

Case 1. The two vertices of Γ_{1} are not parallel.

It is clear that any family of (n_{2}/2) + 1 parallel positive edges contain a Scharlemann cycle {e_{1}, e_{2}}. Moreover, they must lie on one side of the family, for otherwise there would be an extended Scharlemann cycle. Since n_{2} > 2, the Scharlemann cycles can not be on the side near the boundary of P_{1}. Also, by Lemma 5.2(3) we may assume that both Scharlemann cycles have the same label pair, say {1, 2}. It is now clear that the labeling of the graph looks like that in Figure 5.1(c). In particular, for each label j there is a negative edge in Γ_{1} with ends labeled j and j + 1. By the parity rule, v_{j} and v_{j+1} are parallel. Thus all vertices of Γ_{2} are parallel. But then there can not be any positive edges in Γ_{1}, contradicting the existence of family a and d.

Case 2. The two vertices of Γ_{1} are parallel.

Since the families of edges in a and d contain Scharlemann cycles, by Lemma 5.2(5) F_{2} is a separating sphere. Hence n_{2} is even. In this case all edges on Γ_{1} are positive, so b and c also represents at most (n_{2}/2) + 1 edges, and we have 4[(n_{2}/2) + 1] ≥ 3n_{2}. Since n_{2} is even, and n_{2} > 2, this holds only if n_{2} = 4, and each family contains 3 edges. Thus the graph looks like that in Figure 5.1(c). One can see that there is an edge with both ends labeled 4, which is impossible by the parity rule. □

Figure 5.1

Lemma 5.4. (1) If Γ_{2} has a Scharlemann cycle then F_{1} is a separating annulus. (2) Γ_{2} can not have two Scharlemann cycles with different label pairs.

(3) Γ_{2} can not have an extended Scharlemann cycle.

Proof. (1) Let {e_{1}, …, e_{n}} be a Scharlemann cycle in Γ_{2} with label pair {1, 2}, bounding a Scharlemann disk D on the surface F_{2}. Let V be the part of the Dehn filling solid torus J_{1} in M(γ_{1}) lying between the two meridian disks u_{1}, u_{2}, and containing no other u_{j}. If F_{1} is a nonseparating annulus, then the frontier of N(F_{1} ∪ V ∪ D) in M(γ_{1}) consists of two nonseparating annuli F_{1} and F′_{1}. The annulus F′_{1} has 2 less intersection with J_{1} than F_{1}. Since M(γ_{1}) is irreducible and ∂-irreducible, any nonseparating annulus is essential. Hence F′_{1} is essential in M(γ_{1}), contradicting the minimality of n_{1}.

(2) On the annulus F_{1} consider the subgraph G = e_{1} ∪ … ∪ e_{k} ∪ u_{1} ∪ u_{2} of Γ_{1}. If G is contained in a disk D_{1}, then as in the proof of [CGLS, Lemma 2.5.2] it is easy to see that a regular neighborhood of D_{1} ∪ V ∪ D is a once punctured lens space. Since we have assumed that M(γ_{1}) is irreducible, this is impossible. Hence we may assume that G cuts F_{1} into two annuli A_{1}, A_{2} and some disk components. Let A′_{i} be the closure of F_{1} − A_{i}. Consider the manifold Y = N(A′_{1} ∪ V ∪ D). Clearly, ∂Y is a torus, so the frontier of Y in M(γ_{1}) is an annulus Q.

Claim. Q is an essential annulus in M(γ_{1}).

Proof. The central curve C of Q is isotopic to the central curve C′ of F_{1}. Since F_{1} is incompressible, C′ (and hence C) is not null homotopic in M(γ_{1}), so Q is also incompressible. If Q is not essential, it has a boundary compressing disk D′. Put X = M(γ_{1}) − IntY. We have assumed that M(γ_{1}) is irreducible, so if X contains D′, then it is a solid torus with Q as a longitudinal annulus. Similarly for Y.

First assume D′ is in X. Notice that F_{1} can be isotoped into X. Choose D′ so that D′ ∩ F_{1} is minimal. If D′ ∩ F_{1} = ∅, then F_{1} would lie in a 3-ball. If D′ ∩ F_{1} ≠ ∅, an outermost component of D′ − F_{1} disjoint from Q would be a boundary compressing disk of F_{1}. Both cases contradict the essentiality of F_{1}.

Now assume D′ is in Y. Then Y is a solid torus with Q a longitudinal annulus. Let Ŷ be the manifold obtained by attaching a 2-handle H to Y along the longitudinal annulus ∂Y − Q. Then Ŷ is a 3-ball. Recall that Y = N(A′_{1} ∪ V ∪ D), so it can be considered as obtained by attaching a 2-handle H′ = N(D) to the handlebody W = N(A′_{1} ∪ V) along the curve ∂D. Switch the order of the two 2-handle additions. It is easy to see that after attaching H to W, the manifold is a solid torus, with ∂D intersecting a meridian exactly k times, where k > 1 is the length of the Scharlemann cycle. Therefore Ŷ is a punctured lens space, which is a contradiction. □

We continue with the proof of Lemma 5.4. Notice that if A′_{1} contains m vertices of Γ_{1} (including u_{1} and u_{2}), then the new essential annulus Q above intersects J_{1} exactly 2m − 2 times. By the minimality of n_{1} we must have 2m − 2 ≥ n_{1}.

Suppose {e′_{1}, …, e′_{t}} is another Scharlemann cycle of Γ_{2} with label pair {p, p+1}. Since the label pairs are different, they can have at most one label in common, say p = 2. Let G′ = e′_{1} ∪…∪e′_{t} ∪u_{p} ∪u_{p+1} be the corresponding graph on F_{1}. The two graphs G′ and G are disjoint except possibly intersecting at their common vertex u_{p} = u_{2}. Like before, G′ can not be contained in a disk, hence we may assume that G′ is contained in the annulus A_{1} ∪ D_{2}.

By the above, the annulus A_{1} contains at most n_{1}/2 − 1 vertices, so the annulus A_{1} ∪ D_{2} contains at most n_{1}/2 vertices. Applying the Claim to G′, we see that the frontier of Y′ = N((A_{1} ∪ D_{2}) ∪ V′ ∪ D′) is an essential annulus in M(γ_{1}) intersecting J_{1} at most 2(n_{1}/2)−2 = n_{1} −2 times. This contradicts the minimality of n_{1}, completing the proof of (2).

(3) Let {e_{1}, e_{2}, e_{3}, e_{4}} be an extended Scharlemann cycle with label pair {2, 3}, say. Then as above, the set C_{1} = e_{2} ∪ e_{3} ∪ u_{2} ∪ u_{3} cuts F_{1} into two annuli, each containing exactly n_{1}/2 − 1 vertices of Γ_{1} in its interior. The cycle C_{2} = e_{1} ∪ e_{4} ∪ u_{1} ∪ u_{4} must lie on one of these two annuli. Like C_{1}, the cycle C_{2} can not be contained in a disk, so F_{1} − C_{2} consists of two annuli. Let A_{1} be the one which does not contain C_{1}. Let D be the disk on F_{2} bounded by e_{1}, e_{4} and two arcs on the boundary of fat vertices. Let V be the part of the Dehn filling solid torus J_{1} between u_{1} and u_{4} and which contains u_{2} and u_{3}. By the same proof as in (2), one can show that the frontier of N(A_{1} ∪ u_{1} ∪ u_{2} ∪ V ∪ D) is an essential annulus in M(γ_{1}) which intersects J_{1} less than n_{1} times, leading to a contradiction to the minimality of n_{1}. □

Consider a disk F_{2} in the interior of M(γ_{2}). We assume that ∂F_{2} is disjoint from J_{2}. Recall that F_{1} ∩ F_{2} form the graph Γ_{2} in F_{2} which may have some ghost edges connecting the fat vertices of Γ_{2} to ∂F_{2}.

Definition 5.5. (1) A disk F_{2} in M(γ_{2}) is a generalized Gabai disk if all the fat vertices on F_{2} are parallel, and the number of ghost edges is less than Δn_{1}, the valency of fat vertices in Γ_{2}.

Lemma 5.6. If Δ = Δ(γ_{1}, γ_{2}) ≥ 3, then M(γ_{2}) can not have a generalized Gabai disk.

Proof. Recall that a non ghost edge of Γ_{2} is an i-edge if it has i as a label on one of its ends. An i-edge cycle in Γ_{2} is a cycle consisting of i-edges.

Since there are less than Δn_{1} ghost edges, at least one of the labels, say i, has the property that there are at most Δ − 1 ghost edges with label i on their non ghost ends, so every fat vertex v_{j} has a non ghost edge with label i at v_{j}. One can then find a cycle of edges in Γ_{2}, each starting with the label i. Such a cycle is called a great i-cycle in [CGLS]. By [CGLS, Lemma 2.6.2] this implies that Γ_{2} has a Scharlemann cycle, so by Lemma 5.4(1) F_{1} is a separating surface. According to Lemma 2.2 of [GLu], any disk D on F_{2} bounded by an i-edge cycle in Γ_{2} contains a Scharlemann cycle if IntD contains no vertices of Γ_{2}. We are done by Lemma 5.4(2) unless all of these Scharlemann cycles have the same label pair {1, 2}, say.

Consider the subgraph Γ′_{2} of Γ_{2} consisting of all i-edges. We may assume that Γ′_{2} is connected, otherwise consider a smaller disk and follow the argument here. There are at least Δn_{2} −(Δ−1) i-edges, with n_{2} vertices. By calculating the Enler characteristic we see that there are at least 1+Δn_{2} −(Δ−1)−n_{2} faces. Since each of them contains at least one Scharlemann cycle, which contains at least 2 edges, there are at least 2(Δn_{2} − Δ − n_{2} + 2) edges in Γ_{1} connecting u_{1} to u_{2}. Since the valency of u_{i} is Δn_{2}, we have

2(Δn_{2} − Δ − n_{2} + 2) ≤ Δn_{2},

i.e., (Δ − 2)(n_{2} − 2) ≤ 0. Since Δ > 2, this holds only if n_{2} ≤ 2. Recall that there is no trivial loops in Γ_{2}, so n_{2} ≠ 1, otherwise all edges would be ghost edges, contradicting the assumption that F_{2} is an generalized Gabai disk. If n_{2} = 2, all non ghost edges must be parallel, and there are more than 3n_{1}/2 such. This number is greater than n_{1}/2+2, so there exists an extended Scharlemann cycle by the proof of [Wu1, Lemma 2.2], which contradicts Lemma 5.4(3). □

Lemma 5.7. If Δ ≥ 3, then M(γ_{2}) contains a generalized gabai disk.

Proof. We will use sutured manifold theory to prove this lemma. One is referred to [Ga1–3] and [Sch1] for definitions and theorems about sutured manifolds. In particular, we will use the planar surface P_{1} = F_{1} ∩ M as a parametrizing surface, and use Theorem 7.8 of [Sch1].

Consider the sutured manifold (X, γ, β), where X = M(γ_{2}), the suture set γ is empty, and β is the knot which is the center of the Dehn filling solid torus J_{2} in X. Let ∂X = ∂_{+}X, which is denoted by R_{+} in [Sch1]. Since X − β is irreducible and ∂-irreducible, and the norm of ∂_{+}X is 0, by definition X is a β-taut sutured manifold.

Recall that a proper surface Q in M = X − IntN(β) is a parametrizing surface if no component of Q is a disk with boundary in ∂_{±}X. By extending over N(β), Q can also be considered as an immersed surface in X with interior embedded in X, and with boundary on ∂X ∪ β. Isotop Q so that ∂Q intersects γ and ∂N(β) in essential arcs or circles. The index of Q is defined as

I(Q) = μ + ν − 2χ(Q),

where μ and ν are the number of essential arcs of ∂Q in γ and ∂N(β) respectively. The index is additive over components of Q, and Q being a parametrizing surface means that the index of each component is nonnegative. If we view γ and ∂N(β) as cusps, then Q would be a surface with some cusp points on its boundary, and μ + ν is exactly the number of cusps on ∂Q. Hence I(Q) = −2χ_{c}(Q), where χ_{c} is the cusped Euler characteristic defined in Section 1.

Take Q = F_{1} ∩ M, the punctured essential annuli in M. Since X has no sutures, and since β is a circle, μ = ν = 0, so the index of Q is I(Q) = −2χ(Q) = 2n_{1}, where as before n_{1} denotes the number of times F_{1} intersects the Dehn filling solid torus J_{1}. We refer the reader to [Sch1] for the definition of taut sutured manifold decomposition, decomposition that respect a parametrizing surface, and sutured manifold hierarchy. An important fact about parametrizing surface is that if a decomposition respect a parametrizing surface, then I(Q′) ≤ I(Q), where Q′ is the parametrizing surface after the decomposition. The following is one of the fundamental theorems in sutured manifold theory.

Theorem [Sch1, 7.8]. If Q is a parametrizing surface for the β-taut sutured manifold (X, γ), then there is a β-taut sutured manifold hierarchy for (X, γ) which respects Q. □

Applying the theorem to (X, γ, β), with Q = F_{1} ∩ M as a parametrizing surface, we get a sutured manifold hierarchy as in the theorem. By definition, each component of ∂M_{n} is a sphere. Denote by Q_{n} = Q ∩ M_{n} the parametrizing surface in M_{n}. Since the hierarchy respects Q, we have I(Q_{n}) ≤ I(Q) = 2n_{1}.

There is a process called cancellation, see [Sch1, Definition 4.1]. If D is a disk component of Q_{n} which passes each of γ_{n} and β_{n} exactly once, then we can cut along D to reduce the number of components in β_{n}, the resulting sutured manifold is still β-taut [Sch1, Lemma 4.3], and the index of the parametrizing surface unchanged. After cancelling all possible components in β_{n}, we get a new set β′_{n}, for which Q_{n} has no cancellation disks.

If β′_{n} = ∅, then (M_{n}, γ_{n}) would be ∅-taut, so by Corollary 3.9 of [Sch1], the original manifold X would also be ∅-taut, (see Proof of 9.1 from 9.7 on [Sch1, Page 608] for more details.) This would be a contradiction because X = M(γ_{2}) was assumed reducible. Therefore, there must be a component P of ∂_{+}M_{n} which contains some points of β′_{n}.

There is a graph Γ(P) on P constructed in the obvious way: The vertices are P ∩ J_{2} = P ∩ N(β′_{n}), where J_{2} = N(β) is the Dehn filling solid torus, and the edges are the arcs in P ∩ Q = P ∩ Q_{n}. The orientation of β′_{n} comes from that of the knot β, and from the definition of sutured manifold decomposition we know that β′_{n} always intersect ∂_{+}M_{n} in the same direction. In particular, all the fat vertices of Γ(P) are parallel. Thus if P is a sphere, then by removing a small disk from P, the resulting surface is a generalized Gabai disk for F_{2} with no ghost edges, and we are done.

Therefore we assume that P lies on a sphere S of ∂M_{n} which contains some sutures. If P is not a disk, consider a disk component P′ of S ∩ ∂_{±}M_{n}. If P′ has some intersection with β′_{n}, we can take P′ instead of P. If P′ does not intersect β′_{n}, then the existence of a non disk component in S ∩ ∂_{±}M_{n} implies that some components in S ∩ ∂_{±}M_{n} is compressible in M_{n} − β′_{n}, which contradicts the βtautness of M_{n}. Therefore we may assume that P is a disk in S. Since all fat vertices of Γ(P) are parallel, we need only show that Γ(P) has less than Δn_{1} ghost edges. P would then be the required generalized Gabai disk for F_{1}.

Let W be the component of M_{n} containing S. Let D_{1}, …, D_{k} be the components of Q_{n} in W which intersect ∂P. Thus all ghost edges of Γ(P) are contained in ∪∂D_{i}. Recall that the index of D_{i} is I(D_{i}) = μ_{i} + ν_{i} − 2χ(D_{i}). Since each suture and each arc of β′_{n} connect two components of ∂_{±}M_{n} with different orientation, μ_{i} + ν_{i}, the number of cusps on ∂D_{i}, is always even. Since there is no cancellation disk anymore, either μ_{i} +ν_{i} ≥ 4, or χ(D_{i}) ≤ 0. In any case we have I(D_{i}) ≥ (μ_{i} +ν_{i})/2. Summing over all such disks we have

∑(μ_{i} + ν_{i}) ≤ 2 ∑ I(D_{i}) ≤ 2I(Q_{n}) ≤ 4n_{1}.

The left hand side is equal to the total number of arc components of ∪∂D_{i} on ∂_{±}M_{n}, exactly half of which are on ∂_{+}M_{n}. It follows that the number of ghost edges on Γ(P) is at most 2n_{1}. Hence P is a generalized Gabai disk for F_{1}, completing the proof of Lemma 5.7. □

The contradiction in the conclusion of Lemmas 5.6 and 5.7 completes the proof of Theorem 5.1.

In [GLu2] Gordon and Luecke showed that if M(γ_{1}) = S^{3} and M(γ_{2}) is toroidal then Δ = Δ(γ_{1}, γ_{2}) ≤ 2. Moreover, if Δ = 2 then the essential torus in M(γ_{2}) intersects the Dehn filling solid torus exactly twice.

Question 5.8. If in Theorem 5.1 we have Δ(γ_{1}, γ_{2}) = 2, is it true that n_{1} = 2?

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Department of Mathematics, University of Iowa, Iowa City, IA 52242

   E-mail address: wu@math.uiowa.edu