COMBINATORIAL METHODS IN DEHN SURGERY

C. McA. Gordon

Abstract. This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings M(α) and M(β) on a given 3-manifold M, each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in M enables one to find faces of these graphs which give useful topological information about M(α) and M(β), and hence, in certain cases, good upper bounds on the intersection number Δ(α, β) of the two filling slopes.

1. Introduction

This expository paper is an expanded version of part of the author’s lectures at the Workshop. The remaining part of the lectures covered material that is discussed in Sections 2 and 3 of the survey article [Go].

Here, we give an introductory account of the combinatorial methods developed by John Luecke and the author (mainly in the proof of the Knot Complement Conjecture [GL2]) for studying various questions about Dehn surgery. For more details, see [GL2], [GL3] and [GL4].

The setting is that we have two Dehn fillings M(α), M(β) on a 3-manifold M, which contain surfaces P̂, Q̂ respectively, and the main goal is to obtain sharp upper bounds on the intersection number of α and β, under certain hypotheses on P̂ and Q̂.

In Section 2 we describe how P̂ and Q̂ give rise to labelled intersection graphs G_{P} ⊂ P̂, G_{Q} ⊂ Q̂. The idea of using such graphs in the context of Dehn surgery is due to Litherland [L].

In Section 3 we introduce the algebraic notion of a type, which plays a key role in the theory.

In Section 4 we show how certain faces or collections of faces of G_{P} give topological information about M(β) (and/or Q̂), and of course similarly for G_{Q}.

Typeset by AMS-TEX

Partially supported by NSF Grant DMS-9626550.

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

To appear in the Proceedings of the Workshop of Knots ’96, the fifth MSJ International Research Institute in Knot Theory, Waseda University, Tokyo, Japan, 1996.

Section 5 contains the main combinatorial assertion, Theorem 5.1, which says that, if G_{P} and G_{Q} are graphs as in Section 2, then either G_{P} represents all types or G_{Q} contains a configuration we call a (p − χ(P̂))-web.

Finally, Section 6 contains some applications, mainly to Dehn surgery on knots in S^{3}, although we also say something about the case where M(α) and M(β) are reducible.

We have tried to keep the discussion as general as possible, allowing P̂ and Q̂ to have arbitrary genus until the last part of Section 5. It must be admitted that this generality is somewhat bogus, inasmuch as in all the applications in Section 6, Q̂ is a 2-sphere and P̂ is either a 2-sphere or a torus. (For statements which apply to surfaces of arbitrary genus see [T] and [R].) The main point is that if Q̂ is a 2-sphere then, if G_{Q} contains a (p − χ(P̂))-web, it contains one that is innermost in the obvious sense, and in such a great web one can find useful faces, namely Scharlemann cycles. Nevertheless, we hope that presenting the material in this way clarifies the logical structure of the argument.

2. Graphs of surface intersections

Let M be a compact, orientable, irreducible 3-manifold with torus boundary. Let α be a slope, that is, the isotopy class of an essential embedded circle, on ∂M. Then the result of α-Dehn filling on M is the manifold M(α) = M ∪ V_{α}, where V_{α} is a solid torus, glued to M via a homeomorphism ∂M → ∂V_{α} taking α to the boundary of a meridian disk of V_{α}.

If α and β are two slopes on ∂M, then Δ(α, β) will denote the minimal geometric intersection number of α and β.

One finds in practice that statements of the following form tend to hold, and this is the main type of result that we shall be seeking:

if M(α) and M(β) have certain special properties, then Δ(α, β) is bounded above by some Δ_{0}.

The bound Δ_{0} will depend on the properties in question. The situation in which we are particularly interested, and to which our methods apply, is when M(α) and M(β) contain certain kinds of surfaces.

So suppose that P̂, Q̂ are (closed, orientable) surfaces in M(α), M(β) respectively. We may assume that P̂ meets V_{α} in a finite collection of meridian disks, so that P = P̂ ∩ M is a surface in M each of whose boundary components has slope α. Similarly, Q̂ gives rise to a surface Q in M whose boundary components have slope β.

For our machinery to be applicable, it is important that ∂P and ∂Q be nonempty. So, we may state our basic assumption as follows:

(A1) there are properly embedded surfaces (P, ∂P), (Q, ∂Q) ⊂ (M, ∂M) such that ∂P and ∂Q are non-empty, and each component of ∂P (resp. ∂Q) has slope α (resp. β).

By an isotopy of (say) P we may assume in addition that

(A2) P and Q intersect transversely, and each component of ∂P meets each component of ∂Q in exactly Δ(α, β) points.

It is clear, however, that for any α and β we can always find surfaces P and Q, of arbitrary genus, satisfying (A1) and (A2), and therefore we must impose some additional conditions if we are to extract any non-trivial information from this situation. The technical assumption which enables us to do this is:

(A3) each arc component of P ∩ Q is essential in P and in Q.

By (A2), P ∩Q consists of a finite disjoint union of circles and properly embedded arcs, the endpoints of the arcs being the points of intersection of ∂P with ∂Q. Condition (A3) says that for no arc component γ of P ∩ Q is there an arc δ ⊂ ∂P, with ∂γ = ∂δ, and a disk D ⊂ P such that ∂D = γ ∪ δ (and similarly for Q).

We now describe some conditions under which (A3) holds.

Recall that a closed orientable surface S in a 3-manifold N is essential if either S has positive genus, is incompressible in N, and is not parallel to a component of ∂N, or S is a 2-sphere which does not bound a 3-ball in N.

First suppose

(a) P̂ is essential in M(α) and M contains no essential surface homeomorphic to P̂.

Let K_{α} ⊂ M(α) denote the core of the solid torus V_{α}. We choose P̂ (among all essential surfaces in M(α) homeomorphic to P̂) so that p = |∂P | = |P̂ ∩ K_{α}| is minimal. Then p > 0 by hypothesis. Also, standard arguments show that P is incompressible and boundary incompressible in M. In particular, we may assume that no circle component of P ∩ Q bounds a disk in Q.

If Q̂ also satisfies (a) (with M(α) replaced by M(β)), and is chosen to minimize q = |∂Q| = |Q̂ ∩ K_{β}|, then, again by standard arguments, we may assume that condition (A3) holds.

Another case of interest is

(b) P̂ is a Heegaard surface for M(α) and K_{α} cannot be isotoped to lie on P̂.

If P̂ satisfies (a) and Q̂ satisfies (b) (for M(β)), then Gabai [Ga2] shows that (A3) can be achieved if K_{β} is put in thin position with respect to Q̂. (Gabai explicitly treats the case when M(β) ≅ S^{3} and Q̂ ≅ S^{2}, but his argument carries over verbatim to any Heegaard surface Q̂.)

Finally, if P̂ and Q̂ both satisfy (b) (for M(α) and M(β) respectively), then Rieck has shown [R], using thin position, that again (A3) can be assumed to hold. (The case M(α) ≅ M(β) ≅ S^{3}, P̂, Q̂ ≅ S^{2} is done in [GL2], and was also proved independently by Gabai (unpublished).)

Given surfaces P, Q in M satisfying (A1), (A2) and (A3), we focus on the arc components of P ∩ Q, as they lie in P and in Q. We regard these arcs as defining graphs G_{P} and G_{Q} in P̂ and Q̂ respectively, in the obvious way. Thus the (fat) vertices of G_{P} are the disks P̂ − int P, the edges of G_{P} are the arc components of P ∩ Q as they lie in P, and similarly for G_{Q}.

We now encode some additional structure, as follows. Number the components of ∂P 1, 2, …, p in the order in which they appear on ∂M, and similarly number the components of ∂Q 1, 2, …, q. This gives a corresponding numbering of the vertices of G_{P} and G_{Q}. Consider an arc component of P ∩ Q. Its endpoints are points of ∂P ∩ ∂Q, the intersections of (say) the ith component of ∂P with the jth component of ∂Q, and the i′th component of ∂P with the j′th component of ∂Q. We then label the endpoints of the corresponding edge of G_{P} with j and j′ at vertices i and i′ respectively, and similarly for G_{Q}. See Figure 1.

Figure 1

We next give a sign, + or −, to each vertex of G_{P}, according to the direction on ∂M of the orientation of the corresponding component of ∂P, induced by some chosen orientation of P. Equivalently, this is the sign of the corresponding intersection point of K_{α} with P̂ (with respect to some chosen orientations of M, P̂ and K_{α}). In particular, if P̂ separates M(α) then the signs of the vertices 1, 2, …, p of G_{P} alternate. Similarly, we give a sign to each vertex of G_{Q}. Note that around each vertex of G_{P} the edge-endpoint labels occur in order 1, 2, …, q, 1, 2, …, q, …, repeated Δ = Δ(α, β) times, the ordering being (say) anticlockwise at a positive vertex and clockwise at a negative vertex. See Figure 2. Similarly, around each vertex of G_{Q} we see the labels 1, 2, …, p, 1, 2, …, p, …, repeated Δ times.

Figure 2

Since M, P and Q are orientable, an arc component of P ∩ Q joins points of intersection of ∂P with ∂Q of opposite sign. Hence we have the parity rule: if an edge of G_{P} joins vertices i and i′ and the corresponding edge of G_{Q} joins vertices j and j′, then i and i′ have the same sign if and only if j and j′ have opposite signs. (See Figure 1.)

Condition (A3) becomes, in graph-theoretic language, the condition that G_{P} and G_{Q} contain no trivial loops.

We will denote by q the set of (edge-endpoint) labels {1, 2, …, q} of G_{P}. We also have the associated q-intervals (1, 2), (2, 3), …, (q − 1, q), (q, 1).

Figure 3

Note that if D is a disk face of G_{P}, then ∂D consists of an alternating sequence of edges and corners, where the edges are edges of G_{P} (i.e., arc components of P ∩Q), and the corners are q-intervals (i, i + 1) (i.e., arcs in ∂P between consecutively labelled components i, i + 1 of ∂Q. See Figure 3.

Similar remarks apply to G_{Q}.

We may describe the philosophy behind our approach as consisting of two parts:

Combinatorial: use the one-one correspondence between the edges of the labeled graphs G_{P} and G_{Q} to show that either G_{P} contains certain configurations or G_{Q} contains certain (perhaps other) configurations. Topological: use the existence of certain collections of faces of G_{P} (resp. G_{Q}) to get topological information about the pair (M(β), Q̂) (resp. (M(α), P̂)).

These will be elaborated in Sections 5 and 4 respectively.

Remark. For simplicity we have assumed that ∂M is a torus, but everything carries over without much change if we allow M to have additional boundary components (with α and β being slopes on some fixed torus component of ∂M). In this more general setting, we may also allow P̂ and/or Q̂ to have non-empty boundary.

3. Types

We now introduce a purely algebraic concept, that of a type, which plays an important role in the theory under discussion. The reader may choose to skip to Section 4 and return to the present section when necessary.

Let q be a positive integer. A q-type is a q-tuple (ε_{1}, ε_{2}, …, ε_{q}), where ε_{i} = ±, 1 ≤ i ≤ q.

We say that a = (a_{1}, a_{2}, …, a_{q}) ∈ ℤ^{q} represents the q-type τ = (ε_{1}, ε_{2}, …, ε_{q}) if and only if

(1) for some η = ±, sign a_{i} = ηε_{i} for each i such that a_{i} ≠ 0; and

(2) ∑_{i=1}^{q} |a_{i}| ≥ 2.

Condition (2) merely excludes the zero vector, together with vectors all of whose co-ordinates are zero except one, which is ±1.

Note that a represents τ if and only if it represents −τ. Also, there are 2^{q−1} q-types up to sign, including the trivial type, which by definition has ε_{1} = ε_{2} = ⋯ = ε_{q}.

Example. There are 2^{2} = 4 3-types (up to sign): +++, −++, +−+, and ++−. (1, 0, −2) represents − + + and + + −.

A set A ⊂ ℤ^{q} represents all q-types if and only if, for each q-type τ, there exists a ∈ A such that a represents τ.

Examples.

(1) If |k| ≥ 2, then {(0, …, 0, k, 0, …, 0)} represents all q-types.

(2) {(1, 0, 1), (−1, 1, 0), (1, 1, −1)} represents all 3-types.

(3) If 1 ≤ r ≤ q, then A ⊂ ℤ^{q} represents all q-types if and only if {(a_{1}, …, a_{r}, 0, a_{r+1}, …, a_{q}) : (a_{1}, …, a_{q}) ∈ A} represents all (q +1)-types.

For A ⊂ ℤ^{q}, let c(A) be the number of co-ordinates 1 ≤ i ≤ q such that a_{i} ≠ 0 for some a ∈ A. Then we may regard A ⊂ ℤ^{c(A)} ⊂ ℤ^{q} in the obvious way. By (3) above, A represents all c(A)-types if and only if it represents all q-types. Thus we may use the phrase A represents all types without ambiguity.

The key property of types is the following “all types implies torsion” theorem due to Parry [P].

Theorem 3.1 (Parry). If A ⊂ ℤ^{q} represents all types then there exists A_{0} ⊂ A such that ℤ^{q} modulo the subgroup generated by A_{0} has non-trivial torsion.

We remark that it is not true that A_{0} can be chosen to represent all types; equivalently, it is not true that if A ⊂ ℤ^{q} is a minimal set of representatives of all types then ℤ^{q}/(A) has non-trivial torsion. (Here, minimal means that no proper subset of A represents all types.) For example, A = {(1, 2, 1), (−1, 1, 1), (1, −1, 2), (1, 1, −1)} is clearly a minimal set of representatives of all types, but ℤ^{3}/(A) = 0.

A useful fact, proved in [GL4, Lemma 4.4], is that if A ⊂ ℤ^{q} represents all types then either A contains a basis for ℝ^{q} or there exists A_{1} ⊂ A such that c(A_{1}) < q and A_{1} represents all types. As an immediate consequence we have

Lemma 3.2. If A is a minimal set of representatives of all types then A contains a basis for ℝ^{c(A)}.

4. The topology

Consider the solid torus V_{β} ⊂ M(β). The surface Q̂ meets V_{β} in q meridian disks, cutting V_{β} into q 3-balls. For each q-interval λ = (i, i + 1), let H_{λ} denote the 3-ball (1-handle) consisting of that part of V_{β} between the meridian disks whose boundaries are components i and i + 1 of ∂Q.

Let D be a disk face of G_{P}. Then D ⊂ M ⊂ M(β), where the edges of ∂D are arcs in Q, and the corners (i, i + 1) of ∂D are arcs in ∂P running along the corresponding 1-handle H_{(i,i+1)}. See Figure 4, which shows ∂D for the face D illustrated in Figure 3.

Figure 4

Let D be a set of disk faces of G_{P}. Let c(D) be the set of q-intervals λ = (i, i+1) that occur as corners of faces in D. Let e(D) be the set of edges of G_{Q} corresponding to edges belonging to faces in D.

Suppose that the edges e(D) are contained in a disk E ⊂ Q̂. (Note that this will necessarily be the case if Q̂ is a 2-sphere.) We take E to also contain the fat vertices of G_{Q} at the endpoints of the edges in e(D).

Recall that if Q̂ is an essential surface, then we may assume that no circle component of P ∩ Q bounds a disk in Q. Hence (int D) ∩ Q̂ = ∅. If Q̂ is a Heegaard surface, this need not hold. Assume for the moment, however, that (int D)∩E = ∅; we shall see later (at the end of the present section) that in the cases of interest this can be achieved after some disk exchanges on int D.

Define a 3-manifold N(E, D) ⊂ M(β) by

N(E, D) = nhd(E ∪ ⋃_{λ∈c(D)} H_{λ} ∪ D) .

Note that N(E, D) has a natural handle decomposition, with a single 0-handle (corresponding to E), the 1-handles H_{λ}, and 2-handles whose cores are the elements of D. Also, if W denotes the handlebody nhd(E ∪ ⋃_{λ∈c(D)} H_{λ}), then we can read off the element of π_{1}(W) (and hence of H_{1}(W)) represented by the boundary of a disk D ∈ D by reading the sequence of corners of D (with sign, given by the sign of the corresponding vertex of G_{P}) as we go around ∂D in some direction.

The idea is to find D such that N(E, D), or perhaps the surface ∂N(E, D), gives topological information about M(β), or the surface Q̂, possibly a contradiction.

The simplest example of this philosophy is when D consists of a single face of G_{P} which is a Scharlemann cycle. This is a disk face D of G_{P} such that all the vertices of D have the same sign, and all the corners of D are the same q-interval λ = (i, i + 1). See Figure 5. Scharlemann cycles first appeared (in a slightly different context) in [S1]. The following theorem, which is contained in [S2, Proof of Proposition 5.6], is immediate; see Figure 6. (Note that vertices i and i + 1 of G_{Q} are of opposite sign, by the parity rule.)

Figure 5

Theorem 4.1. Let D be a Scharlemann cycle in G_{P}, such that the edges e(D) lie in a disk E ⊂ Q̂. Then N(E, D) is a punctured lens space.

Note that π_{1}(N(E, D)) ≅ ℤ_{k}, where k is the length of D, i.e., the number of edges in ∂D.

Figure 6

Let us examine the implications for (M(β), Q̂) of the existence of a Scharlemann cycle in G_{P} as in Theorem 4.1.

(Sa) If Q̂ is a Heegaard surface, we conclude tht M(β) has a lens space summand.

Now suppose that Q̂ is an essential surface. Note that (after a small isotopy of N(E, D)) we may assume that ∂N(E, D) ∩ Q̂ = E. Let E′ be the disk \overline{∂N(E, D) − E}, and let Q̂′ = (Q̂ − E) ∪ E′. Then Q̂′ ≅ Q̂. Also, |Q̂′ ∩ K_{β}| = q − 2, since the two points of intersection of K_{β} with Q̂ corresponding to the labels i and i + 1 that appear in the corners of the Scharlemann cycle D, have been eliminated.

We therefore have the following.

(Sb) If Q̂ is an incompressible surface of positive genus, then Q̂′ is also incom-

pressible. But this is a contradiction, because of the minimality of q. (Sc) If Q̂ is an essential 2-sphere, then M(β) has a lens space summand. In fact,

by the minimality of q, Q̂ decomposes M(β) as M′# (lens space). (Sd) For a variant of (Sc), let us say that a 2-sphere S in a 3-manifold N is

ℚ-essential if S does not bound a ℚ-homology ball in N. (In particular, this is the case if S is non-separating.) It is clear that if Q̂ is ℚ-essential in M(β), then so is Q̂′. Therefore, if we choose Q̂ so that q is minimal over all ℚ-essential 2-spheres in M(β), then the existence of a Scharlemann cycle in G_{P} is a contradiction.

Although Scharlemann cycles are very useful, their existence is not guaranteed in all situations of interest. A more global concept, which gives similar topological conclusions, is that a set of faces representing all types.

A q-type is a q-type τ = (ε_{1}, ε_{2}, …, ε_{q}) (see Section 3), where the ith co-ordinate of τ is formally associated with the ith q-interval (i, i + 1).

Let D be a disk face of G_{P}. By taking the algebraic sum of the corners of D we get an element [D] ∈ ℤ^{q}, the free abelian group on the set q-intervals.

We say that D represents the q-type τ if and only if

(1) [D] represents τ (in the sense of Section 3); and

(2) (coherence condition) no q-interval (i, i+1) appears as a corner of D at two vertices of opposite sign.

As an example, the disk D shown in Figure 3 fails to satisfy the coherence condition.

A set of (disk) faces D of G_{P} represents all types if and only if for each q-type τ there exists D ∈ D such that D represents τ. If such a set of faces exists we say that G_{P} represents all types.

Remark. The role of the coherence condition (2) is twofold. First, it ensures that condition (2) in the algebraic definition of representing a type (see Section 3) is automatically satisfied (since G_{P} has no trivial loops). Second, it makes it possible to search for representatives of types in a more organized fashion, as we shall see in Section 5 (cf. Proposition 5.2).

If D is a face of G_{P} such that small neighborhoods in D of the edges e(D) all lie on the same side of Q̂, we say that D lies locally on that side of Q̂. To get the topological conclusions that we want from a set of representatives of all types, we first need to observe the following (cf. [GL2, Proof of Proposition 3.2, part (A)]).

Lemma 4.2. Let D be a set of faces of G_{P} representing all types. Then there exists D′ ⊂ D such that D′ represents all types and the elements of D′ all lie locally on the same side of Q̂.

Proof. Label the two local sides of Q̂ in M(β), B and W (black and white).

Let λ = (i, i + 1) be a q-interval, and recall the definition of the 1-handle H_{λ} at the beginning of this section. Orient (the core of) H_{λ} from the end corresponding to component i of ∂Q to the end corresponding to component i + 1 of ∂Q. Then we have the partition

{q-intervals} = B ∐ W ∐ BW ∐ WB ,

where

λ ∈ \{ \begin{matrix}B \\ W \\ BW \\ WB \end{matrix} \} if H_{λ} runs from \{ \begin{matrix}B to B \\ W to W \\ B to W \\ W to B \end{matrix} \} .

Suppose the conclusion of the lemma is false. Then, if B ≠ ∅, there exists a B-type τ_{B} such that no member of D represents τ_{B}. Similarly, if W ≠ ∅, there exists a W-type τ_{W} such that no member of D represents τ_{W}. Define a q-type τ by

τ|B = τ_{B} , if B ≠ ∅ ; τ|W = τ_{W} , if W ≠ ∅ ;
τ|λ = + , ∀ λ ∈ BW ; τ|λ = − , ∀ λ ∈ WB .

By hypothesis, there exists D ∈ D such that D represents τ. Note that, for some orientation of ∂D, every corner of D in BW ∐ WB runs from B to W, and hence D can contain no such corner. Therefore either all corners of D belong to B, or all corners belong to W. It follows that either B ≠ ∅ and D represents τ_{B}, or W ≠ ∅ and D represents τ_{W}, contradicting the definitions of τ_{B} and τ_{W}. □

Theorem 4.3. Let D be a set of faces of G_{P} representing all types, such that the edges e(D) lie in a disk E ⊂ Q̂. Then there exists D′ ⊂ D such that the elements of D′ all lie locally on the same side of Q̂, H_{1}(N(E, D′)) is finite and non-zero, and ∂N(E, D′) ≅ S^{2}.

Proof. (cf. [GL4, proof of Proposition 4.5]). We may assume that D is minimal. Hence, by Lemma 4.2, the elements of D all lie locally on the same side of Q̂. Also, by Lemma 3.2, there exists D′ ⊂ D such that {[D] : D ∈ D′} is a basis for ℝ^{c(D)}. Let N′ = N(E, D′). Then it follows that H_{1}(N′) is finite and ∂N′ ≅ S^{2}.

It remains to show that H_{1}(N′) ≠ 0.

Let N = N(E, D) = nhd(N′ ∪ (D − D′)). Since ∂N′ ≅ S^{2}, attaching to N′ the 2-handles whose cores are the elements of D −D′ is equivalent to removing from N′ an equal number of open 3-balls. By Theorem 3.1 there exists D_{0} ⊂ D such that H_{1}(N_{0}) has non-trivial torsion, where N_{0} = N(E, D_{0}). Note that N_{0} ⊂ N ⊂ N′.

Suppose H_{1}(N′) = 0. Then N′ is a homology 3-ball, and the Mayer-Vietoris exact sequence gives H_{1}(∂N_{0}) ≅ H_{1}(N_{0}) ⊕ H_{1}(N′ − N_{0}), contradicting the fact that H_{1}(N_{0}) has non-trivial torsion. Hence H_{1}(N′) ≠ 0, as desired. □

Remark. Since H_{1}(N) ≅ H_{1}(N′), the above proof shows that if D is a minimal set of representatives of all types then H_{1}(N(E, D)) is finite and non-zero. As was noted in Section 3, the analogous statement is not true in the purely algebraic setting.

It follows from Theorem 4.3 that if G_{P} represents all types, and the edges of the relevant faces are contained in a disk in Q̂, then we get conclusions almost identical to (Sa), (Sb), (Sc) and (Sd) above for Scharlemann cycles.

(Ta) If Q̂ is a Heegaard surface, then M(β) has a summand N̂′ with H_{1}(N̂′)

finite and non-zero. (Tb) If Q̂ is an incompressible surface of positive genus, this is a contradiction.

(We get a new surface Q̂′, obtained from Q̂ by a disk exchange, with |Q̂′ ∩ K_{β}| = q − |c(D)|.)

(Tc) If Q̂ is an essential 2-sphere, then Q̂ decomposes M(β) as M′ # N̂′, where

H_{1}(N̂′) is finite and non-zero. (Td) If Q̂ is a ℚ-essential 2-sphere, this is a contradiction. Finally, we return to the assumption, when Q̂, is a Heegaard surface, that

(int D) ∩ E = ∅.

Let D be a collection of faces of G_{P} representing all types (this includes the case where D consists of a single Scharlemann cycle). We may assume that (int D) ∩ E consists of a finite number of disjoint circles and arcs, properly embedded in E. Let γ be such a circle or arc, and let E_{1} and E_{2} be the components of E − γ. There is a corresponding partition

c(D) = C_{1} ∐ C_{2} ∐ C_{12} ,

where λ ∈ C_{i} if the 1-handle H_{λ} has both ends in E_{i}, i = 1, 2, and λ ∈ C_{12} if H_{λ} has one end in E_{1} and one in E_{2}.

Then, exactly as in the proof of Lemma 4.2 (the c(D)-type τ here will be defined by orienting all λ ∈ C_{12} so as to run from (say) E_{1} to E_{2}), there exists D′ ⊂ D such that D′ represents all types and c(D′) ⊂ C_{1} (say). Since we may assume that D is a minimal set of representatives of all types, we conclude that the vertices of G_{Q} corresponding to the labels that appear in the corners c(D) all lie in E_{1}.

If γ is an arc, or if γ is a circle and Ē_{1} is a disk, we replace E by Ē_{1} (moved slightly so as to eliminate the intersection γ with int D). If γ is a circle and Ē_{2} is a disk, we use a standard innermost circle argument to cut-and-paste D, using subdisks of Ē_{2}, so as to reduce (int D) ∩ E. Thus we eventually get a collection of disks D̃, whose boundaries are the same as those in D, such that (int D̃) ∩ E = ∅.

Remark. There are situations when it is useful to consider submanifolds of M(β) of the form nhd(E ∪ ⋃ H_{λ} ∪ D) where E is a subsurface of Q̂ other than a disk. For example, the case where D consists of a Scharlemann cycle and E is an annulus arises in [GL3, Section 3].

5. The combinatorics

To state the main result of this section, Theorem 5.1 below, we need the following definition.

A subgraph Λ of G_{Q} is a k-web (k a non-negative integer) if and only if

(1) all the vertices of Λ have the same sign, and

(2) at most k edge-endpoints of G_{Q} at vertices of Λ are not endpoints of edges of Λ.

See Figure 7 for an example.

Figure 7

In the following theorem, G_{P} and G_{Q} are graphs as in Section 2.

Theorem 5.1. Assume that Δ > 1 − χ(P̂)/p. Then either G_{P} represents all types or G_{Q} contains a (p − χ(P̂))-web.

Most of the remainder of this section will be devoted to a sketch of the proof of Theorem 5.1.

First we deal with the case of the trivial type, which is different from the general case.

Define Λ to be the subgraph of G_{Q} consisting of all edges of G_{Q} joining vertices of the same sign. If some component of Λ is not a (p−χ(P̂))-web, then G_{Q} contains more than (p−χ(P̂)) edges joining vertices of opposite sign. By the parity rule, the corresponding edges of G_{P} join vertices of the same sign. Let Σ be the subgraph of G_{P} consisting of all vertices of G_{P}, and all edges of G_{P} joining vertices of the same sign. Then

χ(P̂) = V − E + ∑χ(f) (summed over all faces f of Σ )
< p − (p − χ(P̂)) + ∑χ(f) .

Hence ∑ χ(f) > 0, implying that Σ has a disk face. This face is then a face of G_{P} whose vertices all have the same sign, and which therefore represents the trivial q-type.

For non-trivial q-types we proceed as follows.

Define a graph Γ ⊂ P̂ by:

• the vertices of Γ consist of the fat vertices (i.e., the vertices of G_{P}), together with dual vertices v(D), one in the interior of each face D of G_{P};

• the edges of Γ join each dual vertex v(D) to the fat vertices in the boundary of the corresponding face D.

See Figure 8.

Figure 8

Now let τ be a q-type. We define the directed graph Γ(τ) ⊂ P̂ to be the graph Γ, with edges oriented according to the following rule. Let e be an edge of Γ, with one endpoint at the fat vertex v, lying in a q-interval (i, i + 1) at that vertex. Then we orient e

inwards at v if (τ|(i, i + 1)) · sign v = +, and outwards at v if (τ|(i, i + 1)) · sign v = −.

As an example, if q = 6 and τ = + − + − −−, then around the vertices of G_{P} the edges of Γ(τ) are oriented as shown in Figure 9.

Figure 9

The reason for defining the orientation on the edges of Γ(τ) in this way is the following, which is immediate from the definition.

Proposition 5.2. A disk face D of G_{P} represents τ if and only if the corresponding dual vertex v(D) is a sink or source of Γ(τ).

Our search for representatives of types is thus translated into the study of the directed graphs Γ(τ) ⊂ P̂. Here we will use the following very elegant combinatorial formulation of the Poincaré-Hopf Index Theorem, due to Glass [Gl].

Consider any directed graph Ω in a closed surface S. For each vertex v of Ω, let s(v) be the number of switches (i.e., changes in orientation of successive edges) around v, and for each face f of Ω, let s(f) be the number of switches around ∂f. See Figure 10.

Figure 10

Define the index of a vertex or face by

I(v) = 1 − \frac{s(v)}{2} , I(f) = χ(f) − \frac{s(f)}{2} .

In particular, a vertex of positive index is a sink or source (I(v) = 1), and a face of positive index is a disk whose boundary is a cycle (I(f) = 1).

Lemma 5.3. (Glass). Let Ω be a directed graph in the closed surface S. Then ∑ I(v) + ∑ I(f) = χ(S).

Proof. (See [Gl].) Each corner between adjacent edges at a vertex contributes exactly 1 to ∑ s(v) + ∑ s(f). Hence

χ(S) = V − E + ∑χ(f)

= V − \frac{# corners}{2} + ∑χ(f)

= V − \frac{(∑ s(v) + ∑ s(f))}{2} + ∑χ(f)

= ∑I(v) + ∑I(f) .  □

A label i which at a fat vertex lies immediately between oppositely oriented edges of Γ(τ) is a switch label of τ. (Equivalently, i is a switch label if and only if τ|(i − 1, i) and τ|(i, i + 1) have opposite signs.) If the edges of Γ(τ) adjacent to i are oriented in a clockwise (resp. anticlockwise) direction around i, then i is a clockwise (resp. anticlockwise) switch label. Note that this is well-defined, i.e., independent of the sign of the fat vertex under consideration; see Figure 9.

Let C(τ), A(τ) ⊂ q be the set of clockwise and anticlockwise switch labels of τ, respectively.

As an example, for the type τ illustrated in Figure 9, C(τ) = {2, 4}, A(τ) = {1, 3}.

An edge e of G_{P} is a clockwise (resp. anticlockwise) switch edge if the labels at the endpoints of e both belong to C(τ) (resp. A(τ)). A switch edge is an edge of G_{P} that is either a clockwise or an anticlockwise switch edge.

Note that the faces of Γ(τ) are in one-one correspondence with the edges of G_{P}, and that under this correspondence, faces of Γ(τ) of index 1 correspond to switch edges of G_{P}. See Figure 11.

Figure 11

Now let τ be a non-trival q-type, and write c(τ) = |C(τ)|, a(τ) = |A(τ)|. Thus c(τ) = a(τ) > 0.

If v is a fat vertex of Γ(τ), then

s(v) = Δ(c(τ) + a(τ)) = 2Δc(τ) .

Therefore

I(v) = 1 − \frac{s(v)}{2} = 1 − Δc(τ) .

Assume that G_{P} does not represent τ. Then by Proposition 5.2, ∑_{v dual} I(v) ≤ 0. Therefore, by Lemma 5.3,

∑I(f) ≥ χ(P̂) − ∑_{v fat}I(v) = χ(P̂) − p(1 − Δc(τ)) .

Note that since c(τ) ≥ 1 and Δ > 1 − χ(P̂)/p by hypothesis, the last quantity above is > 0.

By the remark above following the definition of a switch edge, the number of switch edges of G_{P} ≥ ∑ I(f). Hence we may assume without loss of generality that there are at least \frac{∑ I(f)}{2} clockwise switch edges, and so the number of endpoints of clockwise switch edges ≥ ∑ I(f) ≥ pΔc(τ) − (p − χ(P̂)).

Now there are Δc(τ) clockwise switch labels at each vertex of G_{P}, giving a total of pΔc(τ) such labels in G_{P}, and so there are at most (p − χ(P̂)) of these labels that are not endpoints of clockwise switch edges of G_{P}. Let Λ be the subgraph of G_{Q} consisting of those edges that correspond to clockwise switch edges of G_{P}. (In particular, the vertices of Λ correspond to a subset of C(τ).) As noted above, Λ is non-empty. Also, there are at most (p − χ(P̂)) occurrences of labels at vertices of Λ that are not endpoints of edges of Λ. Hence if

all elements of C(τ) have the same sign,  (∗c)
and all elements of A(τ) have the same sign,  (∗a)

then all the vertices of Λ have the same sign, and so (a connected component of) Λ is a (p − χ(P̂))-web, as desired. (Note that we need to make both assumptions (∗c) and (∗a), as we have no control over whether there are at least \frac{∑ I(f)}{2} clockwise or anticlockwise switch edges.)

If at least one of (∗c) and (∗a) does not hold, then we take derived types until both do hold, and then work backwards. We now briefly discuss this inductive procedure.

For this, we have to work in the following more general setting. Consider a non-empty set of labels L ⊂ q. Associated with L is the set of L-intervals, which are the intervals (ℓ_{1}, ℓ_{2}) between successive elements of L. An L-type is defined in the obvious way, to be an |L|-type where the ith co-ordinate is formally associated with the ith L-interval.

Define G(L) to be the subgraph of G_{P} consisting of all edges of G_{P} with at least one endpoint label belonging to L. Thus G(L) and G_{P} have the same vertices, and G(q) = G_{P}.

Note that although the corners of the faces of G(L) are not necessarily Lintervals, each is contained in a unique L-interval. Hence, for each disk face D of G(L), by taking the algebraic sum of the L-intervals containing the corners of D we get an element [D] ∈ ℤ^{|L|}. Then, very much as before, we say that D represents the L-type τ if and only if

(1) [D] represents τ (in the sense of Section 3); and

(2) (coherence condition) no L-interval occurs as the L-interval containing a corner of D at two vertices of D of opposite sign.

We define the directed graph Γ(τ) as before (starting with G(L)), orienting each edge according to the restriction of τ to the L-interval containing the corresponding corner (and the sign of the fat vertex in question). Proposition 5.2 continues to hold with G_{P} replaced by G(L).

Let τ be a non-trivial L-type. We have C(τ), A(τ) ⊂ L, the set of clockwise (resp. anticlockwise) switches of τ, as before. The argument we have just given above carries over verbatim to our present more general setting, so we have the following lemma.

Lemma 5.4. Let τ be a non-trivial L-type, satisfying (∗c) and (∗a), such that G(L) does not represent τ. Then G_{Q} contains a (p − χ(P̂))-web.

The derivative of a non-trivial L-type τ is the C(τ)-type dτ defined as follows. Let (c, c′) be a C(τ)-interval. Then (c, c′) contains a unique element a ∈ A(τ). We define dτ by

dτ|(c, c′) = sign a .

Figure 12

Example. Taking q = 16 and starting with the q-type τ = + − + + − + − − + − + + + − −+, Figure 12 below illustrates how one obtains dτ and d^{2}τ.

The key property of the definition of the derivative of a type is the following proposition, which says that to find a representative of τ it is enough to find a representative of dτ.

Proposition 5.5. Let τ be a non-trivial L-type. Then any face of G(C(τ)) representing dτ contains a face of G(L) representing τ.

Sketch Proof. We shall give enough of the proof to at least explain the reason for the definition of dτ.

Let D be a face of G(C(τ)) representing dτ. Let Γ_{D}(τ) = Γ(τ) ∩ D.

Claim. Γ_{D}(τ) has either a face of index 1 or a dual vertex of index 1.

Proof of Claim. This is proved by doubling (D, Γ_{D}(τ)) along the boundary, and applying Lemma 5.3 to the double 2Γ_{D}(τ) ⊂ 2D ≅ S^{2}. Note that the faces of 2Γ_{D}(τ) are the faces of Γ_{D}(τ) (each appearing twice), together with a face f_{e} for each edge e ⊂ ∂D (with e ⊂ f_{e}). Hence the conclusion will follow if we show that

|{fat vertices of 2Γ_{D}(τ) of index 1}| ≤ |{faces f_{e} of index −1}| .

But this in turn follows from the fact that, by definition of G(C(τ)), each e ⊂ ∂D has at least one endpoint label ∈ C(τ) (see [GL2, p.398] for details).

A dual vertex of Γ_{D}(τ) of index 1 will correspond to a face of G(L), contained in D, representing τ. So, by the claim above, we are done once we show that no face of Γ_{D}(τ) has index 1. Such a face corresponds to a switch edge e of G(L). Now e cannot be a clockwise switch edge, for such edges are edges of G(C(τ)), and hence cannot lie in the interior of the face D of G(C(τ)). So assume that e is an anticlockwise switch edge, with endpoint labels a_{1}, a_{2} ∈ A(τ) at vertices v_{1}, v_{2} respectively; see Figure 13. Then, by the parity rule, (sign v_{1})(sign a_{1}) ≠ (sign v_{2})(sign a_{2}). But, recalling the definition of dτ, this contradicts the fact that D represents dτ. □

Figure 13

Returning to the proof of Theorem 5.1, suppose that one of the conditions (∗c), (∗a) fails for the q-type τ. Note that C(τ) = A(−τ). Hence, if we define

τ_{0} = \{ \begin{matrix}τ , & if τ does not satisfy (∗a) \\ −τ , & if τ satisfies (∗a) but not (∗c), \end{matrix}

then (∗a) fails for τ_{0}. Hence dτ_{0} is a non-trivial type. If dτ_{0} satisfies (∗c) and (∗a), stop. If not, define

τ_{1} = \{ \begin{matrix}dτ_{0} , & if dτ_{0} does not satisfy (∗a) \\ −dτ_{0} , & if dτ_{0} satisfies (∗a) but not (∗c) \end{matrix}

Continuing in this way, we obtain a sequence τ_{0}, τ_{1}, …, τ_{n}, n ≥ 1, where τ_{0} = ±τ, τ_{i} = ±dτ_{i−1} is a non-trivial C(τ_{i−1})-type, 1 ≤ i ≤ n, and τ_{n} satisfies (∗c) and (∗a).

If G(C(τ_{n−1})) does not represent τ_{n}, then, by Lemma 5.4, G_{Q} contains a (p − χ(P̂))-web.

If G(C(τ_{n−1})) does represent τ_{n}, then successive applications of Proposition 5.5 show that G_{P} represents τ.

This completes our sketch of the proof of Theorem 5.1.

We have seen, in Section 4, that if G_{P} represents all types then we get useful topological information about (M(β), Q̂). What if G_{Q} contains a (p − χ(P̂))-web? One idea is that a web might give rise to Scharlemann cycles in G_{Q} (which then in turn give useful topological information about (M(α), P̂)). In order to make this work, however, we must impose a further condition. Namely, we define a great k-web in G_{Q} to be a k-web Λ which is contained in the interior of a disk D_{Λ} ⊂ Q̂ with the property that any vertex of G_{Q} lying in D_{Λ} is a vertex of Λ. See Figure 14.

Figure 14

To guarantee that such objects exist, we need to specialize to the case where Q̂ is a 2-sphere. It turns out that one can then prove the following theorem.

Theorem 5.6. Assume that Δ > 1 − χ(P̂)/p, and that Q̂ ≅ S^{2}. Then either G_{P} represents all types or G_{Q} contains a great (p − χ(P̂))-web.

The proof proceeds as follows. By Theorem 5.1, if G_{P} does not represent all types then G_{Q} contains a (p − χ(P̂))-web Λ. Let U be a component of Q̂ − nhd(Λ), and let D be the disk Q̂ − U, containing Λ. If Λ is not a great web, then there are vertices of G_{Q} in D that are not vertices of Λ. These vertices correspond to a non-empty subset L ⊂ q. One then proves by induction on |L| that either G_{Q} contains a great (p − χ(P̂))-web or G(L) represents all L-types. (The last step of the induction, which is the desired statement, is the degenerate case where Λ = ∅, L = q, and D is a disk in Q̂ containing G_{Q}.) This inductive argument is quite subtle, and in particular involves relativizing the notion of the derivative of a type. We refer to [GL3, proof of Theorem 2.5] (which does the case where P̂ is a torus), and thence to [GL2, Section 2], for more details.

6. Applications

In this section we sketch the proofs of some results which use the methods we have discussed. The first three are about Dehn surgery on knots in S^{3}, and the proofs of these will all be based on the following statement, which is an immediate consequence of Theorem 5.6 and (Ta) of Section 4.

Theorem 6.1. Suppose that M(β) ≅ S^{3} and that Q̂ is a Heegaard 2-sphere. If Δ > 1 − χ(P̂)/p then G_{Q} contains a great (p − χ(P̂))-web.

When M(β) ≅ S^{3}, we shall make a slight change of notation, writing K for K_{β}, a non-trivial knot in S^{3}, M_{K} for M, the exterior of K, and μ_{K} for β, the meridian of K.

The following theorem is proved in [GL2]. It implies that knots in S^{3} are determined by their complements.

Theorem 6.2. Let K be a non-trivial knot in S^{3}. If M_{K}(α) ≅ S^{3} then α = μ_{K}.

Proof. Suppose that M_{K}(α) ≅ S^{3}, with Δ(α, μ_{K}) ≥ 1. Let P̂ be a Heegaard 2-sphere in M_{K}(α).

By Theorem 6.1, G_{Q} contains a great (p − 2)-web, Λ. Hence there exists a label (in fact at least two labels) i ∈ p such that every occurrence of i at a vertex of Λ is the endpoint of an edge of Λ. Moreover, since all the vertices of Λ have the same sign, no edge of Λ can have the same label at both endpoints, by the parity rule. Hence, by always leaving a vertex of Λ along an edge of Λ with label i at that vertex, we see that Λ contains an i-cycle, that is, a cycle of edges which can be consistently oriented so that the tail of each edge has label i (and all the vertices in the cycle have the same sign). See Figure 15.

Now it is easy to see that a cycle of this kind which is innermost in the disk D_{Λ} (allowing the label i to be an arbitrary element of p) is a Scharlemann cycle in G_{Q}. Since P̂ is a 2-sphere the edges of this Scharlemann cycle necessarily lie in a disk in P̂. But since P̂ is a Heegaard 2-sphere, this is a contradiction, by (Sa) in Section 4 (here, the roles of P and Q are interchanged). □

Remark. We observed in the above proof that a great (p − 2)-web in G_{Q} always contains a Scharlemann cycle. So we may state

Theorem 6.3. If P̂ and Q̂ are 2-spheres, and Δ ≥ 1, then either G_{P} represents all types or G_{Q} contains a Scharlemann cycle.

If Δ ≥ 2 then the following stronger conclusion holds.

Theorem 6.4. If P̂ and Q̂ are 2-spheres, and Δ ≥ 2, then either G_{P} or G_{Q} contains a Scharlemann cycle.

This is an immediate consequence of [CGLS, Proposition 2.5.6] (which deals with the case where P̂ and Q̂ are disks).

Theorem 6.4 already shows that if M(α) ≅ M(β) ≅ S^{3} then Δ(α, β) ≤ 1, implying that there are at most two inequivalent knots with homeomorphic complements (a fact first proved, by a different argument, in [CGLS]). However, there are examples of graphs G_{P} , G_{Q} with Δ = 1 for which the conclusion of Theorem 6.4 fails, illustrating the necessity of introducing the more general concept of representing all types.

The next application addresses the question of when a reducible manifold can be obtained by Dehn surgery on a knot in S^{3}.

Theorem 6.5. Let K be a non-trivial knot in S^{3}. If M_{K}(α) is reducible then (1) M_{K}(α) has a lens space summand; and

(2) Δ(α, μ_{K}) = 1.

Part (1) was proved in [GL2]; part (2) was proved in [GL1] (by an argument different from that outlined below).

Part (1) implies the Property R Conjecture, proved by Gabai [Ga2].

Corollary 6.6 (Gabai). If K is a non-trivial knot in S^{3} then M_{K}(0) is irreducible.

Part (1) also implies

Corollary 6.7. Any homology 3-sphere obtained by Dehn surgery on a knot in S^{3} is prime.

We remark that Auckly, using results from gauge theory, has shown that there are prime homology 3-spheres which cannot be obtained by Dehn surgery on any knot in S^{3} [A].

The situation described in Theorem 6.5 can occur: if K is a torus knot or cable knot then there is a slope α such that M_{K}(α) is reducible. However, the Cabling Conjecture [GS] asserts that these are the only knots with this property. This is still open.

Sketch Proof of Theorem 6.5. Here we take P̂ to be an essential 2-sphere in M_{K}(α). (Note that M_{K} is irreducible.)

By Theorem 6.1, G_{Q} contains a great (p − 2)-web Λ. As in the proof of Theorem 6.2 above, this implies that G_{Q} contains a Scharlemann cycle, and hence that M_{K}(α) has a lens space summand (see (Sc) in Section 4). This proves (1).

To prove (2), assume that Δ(α, μ_{K}) > 1. Then, for homological reasons, P̂ separates M_{K}(α), and hence p is even.

If p = 2 then P is an annulus, and hence K is a cable of a knot K′ (which may be trivial). Now it is not hard to show (see the first four lines of the proof of Theorem 3 in [GL1]) that if M_{K}(α) is reducible for some α with Δ(α, μ_{K}) > 1 then M_{K′}(α′) is reducible for some α′ with Δ(α′, μ_{K′}) > 1. Hence, by induction it suffices to prove the result when K is not cabled.

So assume p ≥ 4. As in the proof of Theorem 6.2 above, there is a label in p such that each of the Δ occurrences of that label at each vertex of Λ is the endpoint of an edge of Λ. One can then show, using the fact that Δ ≥ 2, that Λ contains Scharlemann cycles of G_{Q} on distinct p-intervals (i, i + 1), (j, j + 1) (see [GL4, Proof of Theorem 2.3]). This allows one to construct a new essential 2-sphere P̂′ ⊂ M_{K}(α) such that |P̂′ ∩ K_{α}| < p (see [GL4, Theorem 2.4]) contradicting the minimality of p. □

For our last application to Dehn surgery on knots in S^{3} we consider the situation where M_{K}(α) contains an incompressible torus. Here, we must assume that M_{K} contains no essential torus, or equivalently, that K is not a satellite knot.

Theorem 6.8 [GL3]. Let K be a knot in S^{3} that is not a satellite knot. If M_{K}(α) contains an incompressible torus then Δ(α, μ_{K}) ≤ 2.

Examples with Δ(α, μ_{K}) = 2 have been constructed by Eudave-Muñoz [E2]. There are many examples with Δ(α, μ_{K}) = 1.

Sketch Proof of Theorem 6.8. Let P̂ be an incompressible torus in M_{K}(α), and assume (for a contradiction) that Δ(α, μ_{K}) ≥ 3. In particular, this implies that P̂ separates M_{K}(α), so that p is even.

If p = 2 then one can show that the knot K is strongly invertible (see [GL3, Section 8]) in which case the theorem is proved by Eudave-Muñoz [E1].

So assume that p ≥ 4. By Theorem 6.1, G_{Q} contains a great p-web Λ. An easy euler characteristic argument (see [GL3, Section 4]) shows that there are at least four labels i ∈ p such that Λ contains an i-bigon; see Figure 16. Now such an ibigon is either a Scharlemann cycle, or contains within it an extended Scharlemann cycle, that is, a Scharlemann cycle flanked by bigons; see Figure 17. However, a topological argument shows that an extended Scharlemann cycle in G_{Q} would give rise to a new incompressible torus P̂′ in M_{K}(α) with |P̂′ ∩ K_{α}| < p, contradicting the minimality of p (see [GL3, Theorem 3.2]). We conclude that Λ contains two Scharlemann cycles D, D′ of length 2 on disjoint p-intervals (i, i+1) and (i′, i′ +1). By (Sb) in Section 4, the edges of neither D nor D′ lie in a disk in P̂, hence each pair of edges forms an essential loop on P̂; see Figure 18. Shrinking the 1-handle H_{(i,i+1)} to its core has the effect of gluing together the two (i, i + 1) corners of D (see Figure 19), giving a Möbius band B. Similarly, D′ gives rise to a Möbius band B′. Since ∂B and ∂B′ are parallel on P̂ we may join B and B′ by an annulus in P̂ to get a Klein bottle F in M_{K}(α). We now replace the torus P̂ by the boundary P̂_{0} of a regular neighborhood of F. (Note that P̂_{0} is incompressible, since M_{K}(α) contains an incompressible torus by hypothesis, and is irreducible by Theorem 6.5(2).) But similar combinatorial arguments, applied to P_{0} ∩ Q, in this case lead to a contradiction (see [GL3, Section 6]). □

Figure 16

Figure 17

Figure 19

Finally, we show how the methods we have discussed give some information about the question of when M(α) and M(β) are reducible. Let us say that a 3-manifold is ℚ-reducible if it contains a ℚ-essential 2-sphere.

Theorem 6.9. (i) If M(α) and M(β) are ℚ-reducible then α = β.

(ii) If M(α) is reducible and M(β) is ℚ-reducible, with α ≠ β, then M(α) has

a lens space summand. (iii) If M(α) and M(β) are reducible, with α ≠ β, then either both M(α) and

M(β) have summands M_{α}, M_{β} with H_{1}(M_{α}) and H_{1}(M_{β}) finite and nonzero, or one of M(α), M(β) has a lens space summand.

Remarks. (1) Part (i) is a special case of a result of Scharlemann [S3] proved using the theory of sutured manifolds. The case where M(α) and M(β) are connected sums of copies of S^{1} × S^{2} was proved by Gabai [Ga1].

(2) Scharlemann [S3] proves that under the hypotheses of (ii), M(β) ≅ S^{1} × S^{2} # W, and M(α) ≅ W′ # W′′, where W, W′ and W′′ are ℚ-homology spheres, and that either W′ is a lens space or H_{1}(W) ≠ 0.

(3) It is shown in [GL4], by an elaboration of the methods discussed here, that if M(α) and M(β) are reducible then Δ(α, β) ≤ 1. A different proof has been given by Boyer and Zhang [BZ].

Proof of Theorem 6.9. (i) Let P̂, Q̂ be ℚ-essential 2-spheres in M(α), M(β) respectively with p and q minimal. By Theorem 6.3, either G_{P} represents all types or G_{Q} contains a Scharlemann cycle. But the first contradicts the minimality of q (see (Td) in Section 4), while the second contradicts the minimality of p (see (Sd) in Section 4).

(ii) Let Q̂ be as in (i), and let P̂ be an essential 2-sphere in M(α). By Theorem 6.3 (and (Td)) G_{Q} contains a Scharlemann cycle. Hence M(α) has a lens space summand.

(iii) Let P̂, Q̂ be essential 2-spheres in M(α), M(β) respectively. The result then follows from Theorem 6.3, applied as stated and also with P and Q interchanged, (together with (Sc) and (Tc) in Section 4). □

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Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082

Current address: MSRI, 1000 Centennial Dr., Berkeley, CA 94720

   E-mail address: gordon@math.utexas.edu