3-MANIFOLDS AS VIEWED FROM THE CURVE COMPLEX

John Hempel

Abstract. A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the complex of curves on a surface and a Heegaard splitting as a pair of subcomplexes generated by the equivalent diagrams. We relate geometric and combinatorial properties of these subcomplexes with topological properties of the manifold and/or the associated splitting. For example we show that for any splitting of a 3-manifold which is Seifert fibered or which contains an essential torus the subcomplexes are at a distance at most two apart in the simplicial distance on the curve complex; whereas there are splittings in which the subcomplexes are arbitrarily far apart. We also give obstructions, computable from a given diagram, to being Seifert fibered or to containing an essential torus.

0. Introduction. Throughout S will denote a closed, connected, oriented surface of genus g ≥ 2. The curve complex of S, denoted C(S), will be the complex whose vertices are the isotopy classes of essential simple closed curves in S, and where distinct vertices x_{0}, x_{1}, …, x_{k} determine a k-simplex of C(S) if they are represented by pairwise disjoint simple closed curves. If we fix a hyperbolic metric on S, then each isotopy class contains a unique geodesic. Moreover two isotopy classes have disjoint representatives if and only if their geodesic representatives are disjoint. We will thus always think of vertices as being geodesics and will use the same notation for a simplex of C(S), the corresponding collection of mutually exclusive simple closed curves in S, and their union as a subset of S.

A simplex X of C(S) determines a compression body

V_{X} = S × [0, 1] ∪_{X×1} 2 − handles ∪ 3 − handles

obtained by attaching 2-handles along the components of X × 1 and filling in any resulting 2-sphere boundary components with 3-cells. S × 0 is called the outer boundary of V_{X} and is naturally identified with S.

A pair X, Y of simplexes of C(S) determine a (Heegaard) splitting

(S; V_{X}, V_{Y}) of a 3-manifold M_{X,Y} = V_{X} ∪_{S} V_{Y}

for which (S; X, Y) is a (Heegaard) diagram.

Typeset by AMS-TEX

Research at MSRI is supported in part by NSF grant DMS-9022140

Our goal is to study (compact, oriented) 3-manifolds and their splittings in terms of the geometry and combinatorics of C(S). We will be primarily interested in the case of closed 3-manifolds (V_{X} and V_{Y} handlebodies).

There is a subcomplex K_{X} ⊂ C(S) consisting of those simplexes X′ (and their faces) with (V_{X′}, S) = (V_{X}, S) (see Lemma 1.2). So the pair K_{X}, K_{Y} of subcomplexes of C(S) describes the different diagrams for a fixed splitting. The major questions: What can one say about a splitting in terms of a representative diagram? What can one say about a 3-manifold in terms of the complexes associated with a given splitting?

Perhaps the most natural thing to consider is the geodesic distance function d, defined on the 0-skeleton of C(S) by d(x, y) = the minimal number of 1-simplexes in a simplicial path joining x to y. So d(x, y) ≤ 1 if and only if x ∩ y = ∅ and d(x, y) ≤ 2 if and only if there is some z with x ∩ z = y ∩ z = ∅ (i.e x ∪ y does not fill S). Higher distances are harder to visualize, but it is known [MM] that C(S) has infinite diameter with respect to d. The significance of the distance function to 3-manifolds begins with:

Observation. The splitting (S; V_{X}, V_{Y}) of a closed, oriented 3-manifold M_{X,Y} is: (1) reducible if and only if d(K_{X}, K_{Y}) = 0, and

(2) weakly reducible if and only if d(K_{X}, K_{Y}) ≤ 1.

Here the distance d(K_{X}, K_{Y}) is the minimal distance between their respective vertices; we call it the distance of the splitting. The above observation is merely a restatement of definitions in terms of the distance on C(S). Its significance lies in the theorems of Haken [H] that a splitting without any cancelling handle pairs is reducible if and only if the corresponding manifold contains an essential 2-sphere and of Casson and Gordon [CG1] that a weakly reducible splitting is either reducible or the corresponding manifold contains an incompressible surface. Since splittings of S^{3} are standard [W], a reducible splitting of an irreducible 3-manifold must have a cancelling pair of handles.

We show in section 3:

Theorem. Let M be a closed oriented 3-manifold which is Seifert fibered or which contains an essential torus. Then any splitting of M is a distance ≤ 2 splitting.

The converse is false. As we observe in section 1 there are many hyperbolic 3manifolds with distance 2 splittings. For example, any Dehn surgery on a 2-bridge knot (most of which are hyperbolic manifolds) has a distance ≤ 2 splitting. This follows from

Theorem. If M is obtained by surgery on a link L in S^{3} then any splitting of M which is derived from a bridge presentation of L is a distance ≤ 2 splitting.

Of course, these splittings need not be irreducible nor of minimal genus.

However it is true that:

Theorem. There are distance n splittings of closed, oriented 3-manifolds for arbitrarily large n.

This is shown in section 2 with an argument supplied by Feng Luo. We also give in section 5 an explicit construction of some distance ≥ 3 splittings. However we are unable to answer:

Question. For each n ≥ 3 are there closed, oriented 3-manifolds which have no irreducible ( or no minimal genus) splittings of distance < n ?

We remark that there are 3-manifolds with inequivalent minimal genus splittings [M] and 3-manifolds with irreducible splittings of different genus [CG2], but any two splittings of a given 3-manifold are stably equivalent [R]. However adding a cancelling pair of handles reduces the distance of the splitting to zero. It is not clear whether distance survives to any sort of meaningful invariant for 3-manifolds.

In section 2 we introduce some estimates on the distance function which allows us to prove

Theorem. diam(K_{X}) = ∞.

Which gives an independent proof that diam(C(S)) = ∞.

This also indicates why the problem is difficult – one can have “simple” splittings represented by diagrams (S; X, Y) with d(X, Y) arbitrarily large. However, we show in sections 3 and 4 that there are obstructions, computable from a fixed diagram, for the corresponding splitting and/or manifold to be reducible, weakly reducible, Seifert fibered, contain an essential torus, or be a distance 2 splitting. Examples of their application are given. They arise from enumerating the “square” regions of S − X ∪ Y according to where the edges lie, and are encoded in a stack intersection matrix. This turns out to be a much more accurate measure of the real complexity of the splitting. This builds on ideas introduced by Casson and Gordon [CG2] and extended by Kobayashi [K] as an obstruction to being weakly reducible. They also provide lower bound estimates for some natural invariants of splittings such as the minimal intersection number between essential disks in the two halfs of the splitting.

Section 6 gives an analysis of all genus two, distance two splittings.

1. Preliminaries. Throughout S will denote a closed, connected, oriented surface of genus g ≥ 2. The geometric intersection number of simple closed curves J_{1}, J_{2} in S is

i(J_{1}, J_{2}) = min{#(J′_{1} ∩ J′_{2}) : J′_{i} isotopic to J_{i}}.

A collection J_{1}, J_{2}, … of simple closed curves will be said to meet efficiently if they are in general position and i(J_{i}, J_{j}) = #(J_{i} ∩ J_{j}) for all i, j. This is equivalent to having no disk D ⊂ S with ∂(D) = a_{i} ∪ a_{j} where a_{i} and a_{j} are arcs in J_{i} and J_{j} respectively.

If N is a codimension one, bicollared submanifold of a manifold M, the result of splitting M along N will be a manifold M^{∗} whose boundary contains disjoint copies N^{+} and N^{−} of N together with a map f : M^{∗} → M which maps M^{∗} − N^{+} ∪ N^{−} homeomorphically onto M − N and maps each of N^{+} and N^{−} homeomorphically onto N.

The curve complex C(S) is the complex whose k-simplexes are the isotopy classes of collections of k + 1 mutually exclusive, pairwise non isotopic, essential simple closed curves in S. dim(C(S)) = 3g − 4: a principal simplex of C(S) is a collection of 3g − 3 simple closed curves which splits S into it pairs of pants (thrice punctured 2-spheres).

We will not distinguish notationally between simple closed curves and their isotopy classes.

If X = (x_{0}, x_{1}, …, x_{k}) is a k-simplex of C(S), we define:

N_{X} = normal closure of {x_{0}, x_{1}, …, x_{k}} in π_{1}(S)

and

V_{X} = S × [0, 1] ∪_{X×1} 2 − handles ∪ 3 − handles .

Then V_{X} is a compression body whose outer boundary, S×0, is naturally identified with S and N_{X} = ker{π_{1}(S) → π_{1}(V_{X})} determines V_{X} up to homeomorphisms which restrict to the identity on S.

A (Heegaard) splitting of a compact, orientable 3-manifold M is a representation of M as the union of two compression bodies intersecting on the outer boundary of each. So a pair X, Y of simplexes of C(S) determines a splitting (S; V_{X}, V_{Y}) of the 3-manifold

M_{X,Y} = V_{X} ∪_{S} V_{Y} .

Every such 3-manifold is represented in this way, but our requirement g ≥ 2 precludes the standard genus zero and one representation of S^{3}, Lens spaces, and S^{2} × S^{1}.

We will be concerned primarily with closed 3-manifolds which will be represented as above with V_{X} and V_{Y} handlebodies. So we say X is a full simplex of C(S) if V_{X} is a handlebody. The following gives equivalent properties.

1.1 Lemma. For X = (x_{0}, x_{1}, …, x_{k}) a simplex of C(S) the following are equivalent:

(i) X is full in C(S),

(ii) Every component of S − X is planar,

(iii) S − X has k − g + 2 components, and

(iv) π_{1}(S)/N_{X} is free of rank g.

proof. The arguement is standard, but it is helpful to note that for any simplex X of C(S) of dimension k that

π_{1}(S)/N_{X} = π_{1}(S_{g_{1}}) ∗ … π_{1}(S_{g_{c}}) ∗ F_{r}

where S − X has c components, r = k − g + 2, and ∑ g_{i} = g − r. □

We call the pair X, Y of simplexes a (Heegaard) diagram for the splitting (S; V_{X}, V_{Y}). It has been more traditional to think of diagrams as being given by the smallest dimension simplex which will determine the corresponding compression body (= g −1 for a handlebody), but we find it convenient to allow superfluous vertices – those which can be omitted without changing the compression body – as opposed to essential vertices which cannot be omitted. Specifically, a vertex x of X will be superfluous if there are distinct components of S − X on opposite sides of x, at least one of which is planar. In fact we find that our theorems provide the strongest results when applied to maximal dimensional simplexes. See comments 4.8.

The following is an easy consequence of a theorem of Luo [L].

1.2 Lemma. Two (3g − 4)-simplexes X, X′ of C(S) determine the same handlebody, (V_{X}, S) = (V_{X′}, S), if and only if there is a sequence X = X_{0}, X_{1}, … X_{n} = X′ of (3g − 4)-simplexes of C(S) such that X_{i−1} ∩ X_{i} is a full (3g − 5)-face of each for i = 1, 2, …, n.

Caution: if we leave off “full” the statement becomes true for any two (3g − 4)- simplexes of C(S) [HT].

So for X a full simplex of C(S)

K_{X} = {X′ : X′ is a face of a full simplex X′′ with V_{X} = V_{X′′}}
= {X′ : N_{X′} ⊂ N_{X}}

Is a connected subcomplex of C(S) whose boundary lies in the non full (3g − 5)- simplexes.

We say that a splitting (S; V_{1}, V_{2}) is reducible (respectively weakly reducible) if there are essential disks D_{i} ⊂ V_{i}, i = 1, 2 with ∂(D_{1}) = ∂(D_{2}) (respectively ∂(D_{1}) ∩ ∂(D_{2}) = ∅). A reducible splitting can be written as a connected sum of lower genus splittings or has a canceling pair of handles. If X and Y are full simplexes and S − X ∪ Y is not simply connected then the splitting is obviously reducible.

A wave relative to X, Y and based in X (Y) is an arc in S whose endpoints lie in the same component of X (Y), whose interior misses X ∪ Y, and which is not parallel to an arc in X (Y). A diagram (S; X, Y) will be called generic if X and Y are full simplexes which meet efficiently, S − X ∪ Y is simply connected, and there are no waves relative to X, Y.

If some wave w, say, based in X lies in a component P of S split along X with at least four boundary components, then we can do surgery along w to replace X by a simplex X′ of the same dimension with V_{X′} = V_{X} and i(X′, Y) < i(X, Y) – or else we discover an obvious reduction. There are two choices for the surgery; one will always give a simplification. If P had only three boundary components, this will not work. But then by Lemma 1.1 X would have a proper, full face to which we could apply the above procedure, if appropriate.

If (S; X, Y) is a generic diagram with, say, dim(X) < 3g − 4, then we can expand X to a simplex X′ with X′, Y generic and dim(X′) = dim(X) + 1 as follows. Split S along X to get a disjoint union of planar surfaces. Collapse the boundary components to vertices and identify the families of parallel arcs of the split open Y to single edges to get a graph with one component each in a disjoint union of 2-spheres. Some component has at least four vertices. Take a tree in this component which does not contain all the vertices and whose removal does not separate the component. The boundary of a regular neighborhood of this tree, pulled back to S, represents a vertex we can add to X to get a simplex X′ with X′, Y generic. Call this operation a generic expansion. Together these observations prove:

1.3 Lemma. A pair of full simplexes of C(S) can either be modified to a generic pair of (3g − 4)-simplexes which determine the same splitting by

(i) taking full faces,

(ii) surgery along a wave, and

(iii) generic expansion

or one will discover an obvious reduction to the associated splitting in the process.

Suppose X and Y are faces, not necessarily full, of simplexes X′ and Y′ of C(S). Then there is a natural inclusion M_{X,Y} ⊂ M_{X′,Y′}. If the components of ∂(M_{X,Y}) are all tori, then the components of M_{X′,Y′} − M_{X,Y} are solid tori and M_{X′,Y′} is obtained by Dehn filling on M_{X,Y}. Moreover every Dehn filling of M_{X,Y} is obtained in this way: the meridians of the filling solid tori can be isotoped to miss the 2handles and then pulled back to curves in S to represent additional vertices for X′ and/or Y′. Clearly d(K_{X′}, K_{Y′}) ≤ d(K_{X}, K_{Y}) ≤ d(X, Y).

Let L be a link in S^{3}. We can always isotope L so that for some 3-ball B ⊂ S^{3} L∩Int(B) is the disjoint union of arcs b_{1}, b_{2}, …, b_{n} which cobound mutually disjoint disks D_{1}, D_{2}, …, D_{n} in B with arcs in ∂(N), and L − ∪b_{i} is the disjoint union of arcs a_{1}, a_{2}, …, a_{n} in ∂(B). This is called an n-bridge presentation of L, and the minimal such n is called the bridge number of L.

We can choose a regular neighborhood N = N(L) so that V = Cl(B − N) is a genus n handlebody and N ∩ ∂(B) is the disjoint union of n disks E_{1}, E_{2}, …, E_{n}, each containing some a_{i}. Then Cl(S^{3} −N) is homeomorphic to the result of adding 2-handles to V along any n − 1 of the curves ∂(E_{1}), ∂(E_{2}), …, ∂(E_{n}). See Figure

So (S; X, Y) is a genus n diagram for a splitting of Cl(S^{3} − N) where S = ∂(V), X = {∂(E_{1}), ∂(E_{2}), …, ∂(E_{n−1}}, and Y = {∂(D_{1} ∩V), ∂(D_{2} ∩V), …, ∂(D_{n} ∩V)}. If M is obtained by Dehn surgery on L then this diagram/splitting extends to a diagram/splitting for M which we say is derived from the bridge presentation of L.

1.4 Theorem. If the closed 3-manifold M is obtained by surgery on a nontrivial link L in S^{3} then any splitting of M derived from a bridge presentation of L is a distance ≤ 2 splitting.

Proof. Take a diagram (S; X, Y) for the link complement as described above. So d(X, Y) is an upper bound for the distance of the derived splitting of M.

But d(x_{1}, y_{1}) ≤ 2; since a meridian m_{2} (for S^{3} − V) dual to y_{2} can be chosen disjoint from x_{1} ∪ y_{1}. □

1.5 Corollary. Each irreducible 3-manifold obtained by surgery on a 2-bridge knot has a distance two genus two splitting.

Proof. A weakly reducible, genus two splitting is easily seen to be reducible; so any irreducible splitting derived from a 2-bridge presentation of the knot will be a distance two splitting. □

2. Distance estimates. In this section we give upper and lower estimates on the distance between two curves in C(S) which will be used in later sections as well as in establishing that K_{X} has infinite diameter.

One can easily construct curves at distance two in C(S) which intersect as much as desired. However the intersection number i(x, y) does provide an upper bound to the distance between curves x and y. To see this note that if one replaces the arc on x between two points of x ∩ y which are adjacent on y by this arc on y one gets a curve x′ which meets x at most once (and so d(x, x′) ≤ 2) and which (for appropriate choice) meets y at most half as much as x does. This provides the basis for an inductive proof of:

2.1 Lemma. For vertices x, y of C(S) with i(x, y) > 0

d(x, y) ≤ 2 + 2log_{2}(i(x, y)).

The opposite bound is based on the observation that intersections between curves which persist on passage to covering spaces have a greater infuuence on their distance. To this end we say that a covering space p : S̃ → S separates simple closed curves x and y in S if there are components x̃ of p^{−1}(x) and ỹ of p^{−1}(y) with x̃ ∩ ỹ = ∅. A finite covering p : S̃ → S is called sub-solvable if p can be factored as a composition of cyclic coverings (regular with cyclic covering group: which may be assumed to have prime degree).

2.2 Definition. For distinct vertices x, y of C(S) we define the covering distance between x and y to be:

cd(x, y) = 1+ min{n :there is a degree 2^{n} sub-solvable covering of S which separates x and y}.

2.3 Lemma. Let x and y be distinct vertices of C(S). Then (i) d(x, y) = 2 if and only if cd(x, y) = 2 (ii) cd(x, y) ≤ d(x, y)

Proof. Suppose d(x, y) = 2 Then x ∩ y ≠ ∅ but some vertex z is disjoint from x ∪ y. We may assume z does not separate S; for otherwise x ∪ y lies in one component of S −z and we could replace z by a non separating curve in the other component. We construct a double cover of S by glueing together two copies of S split open along z. One of these components contains a (homeomorphic) lift of x and the other a lift of y. Thus cd(x, y) = 2.

Conversely, suppose cd(x, y) = 2 and that p : S̃ → S is a double covering separating x and y. Then p^{−1}(x ∪ y) has two components and these must be interchanged by the non-trivial covering transformation. Some boundary component of a small regular neighborhood of p^{−1}(x∪y) projects homeomorphically to an essential simple closed curve in S − (x ∪ y).

For (ii) suppose that d(x, y) = n > 2. Then there are vertices

x = x_{0}, x_{1}, …, x_{n} = y

with d(x_{i−1}, x_{i}) = 1. By part (i) there is a double covering p^{∗} : S^{∗} → S separating x_{0} and x_{2}. So for appropriate components x_{i}^{∗} of p^{∗−1}(x_{i}) we have x_{0}^{∗}∩x_{2}^{∗} = x_{2}^{∗}∩x_{3}^{∗} = ⋯ = ∅. So d(x_{0}^{∗}, x_{n}^{∗}) ≤ n − 1.

By induction there is sub-solvable covering q : S̃ → S^{∗} separating x_{0}^{∗} and x_{n}^{∗} and of degree 2^{m} for some m ≤ n − 2. Then p = p^{∗} ∘ q has degree 2^{m+1} ≤ 2^{n−1}, is sub-solvable, and separates x and y. So cd(x, y) ≤ m + 1 ≤ n = d(x, y). □

2.4 Observation. The inequality in (ii) above is, in general, proper. The difference d(x, y) − cd(x, y) can be made arbitrarily large.

Proof. A double cover of S is determined by a homomorphism of π_{1}(S) to ℤ/2ℤ which in turn is given by the mod 2 intersection number with a fixed curve. The cover in part (i) corresponds to intersection with z. Now if (in (ii)) d(x, y) is very large then many of the curves x_{i} will be homologous (mod 2) to the curve generating the cover, and correspondingly d(x_{i−1}^{∗}, x_{i+1}^{∗}) ≤ 1 (for appropriate lifts). Thus d(x_{0}^{∗}, x_{n}^{∗}) will be considerably less than d(x, y). However we necessarily have cd(x_{0}^{∗}, x_{n}^{∗}) ≥ cd(x, y) − 1. □

2.5 Theorem. If h : S → S is a pseudo-anosov homeomorphism and x and y are vertices of C(S) then

lim_{n→∞} cd(x, h^{n}(y)) = ∞.

Proof. Fix an integer m > 0 and let N be the intersection of all subgroups of index 2^{m} in π_{1}(S). Let p : S̃ → S be the corresponding regular covering space. Then p factors through every degree 2^{m} sub-solvable covering and so any pair of curves which is separated by one of these covers is separated by p.

Now N is characteristic and is preserved by h_{∗}; so h is covered by a pseudoanosov homeomorphism h̃ : S̃ → S̃. Now for any essential simple closed curves z, w in S̃, lim(i(z, h̃^{n}(w)) = ∞ – in fact this property can be taken as a definition of pseudo-anosov (cf [FLP]). Pick a component ỹ of p^{−1}(y). Then there is some n_{0} so that for n > n_{0}, h̃^{n}(ỹ) intersects every component of p^{−1}(x). By regularity p cannot separate x and h^{n}(y) for n > n_{0}. Thus cd(x, h^{n}(y)) > m. □

2.6 Theorem. For X a full simplex of C(S), diameter(K_{X}) = ∞.

Proof. One can find two simple closed curves u, v which bound disks in V_{X}, and so represent elements of N_{X} and such that u ∪ v fills S. (cf Figure 2). The product h of the Dehn twists along u and v is pseudo-anosov [P]. Clearly h(K_{X}) = K_{X}. The conclusion follows from 2.5 and 2.3. □

A proof for the following was shown to me by Feng Luo who attributes it to being implicit in the paper [K2] of Kobayashi.

2.7 Theorem. For any d there are full simplexes X, Y of C(S) with d(K_{X}, K_{Y}) ≥ d.

Proof. We regard simple closed curves, with the counting measure, as elements of the space ML(S) (≅ ℝ^{6g−6}) of measured laminations on S and so elements of the space P L(S) (≅ S^{6g−7}) of projective measured laminations on S.

Let X be any full simplex of C(S). It is known [Ma] that the closure, C, in P L(S) of the set of vertices of K_{X} is nowhere dense in P L(S). So there is a pseudo-anosov homeomorphism h : S → S whose stable lamination L is not in C. We claim that

lim_{n→∞} d(K_{X}, h^{n}(K_{X})) = ∞.

To establish this claim it suffices to show that there do not exist sequences

x_{1}, x_{2}, x_{3}, … and y_{1}, y_{2}, y_{3}, …

of vertices of K_{X} with d(x_{n}, h^{n}(y_{n})) bounded.

If not then for some m there are sequences

x_{1}^{j}, x_{2}^{j}, x_{3}^{j}, …; j = 1, 2, …, m

of simple closed curves with x_{n}^{1} = x_{n}, x_{n}^{m} = h^{n}(y_{n}), and x_{n}^{j} ∩ x_{n}^{j+1} = ∅ for all n and for j = 1, 2, …, m − 1.

Now i extends to a continuous function i : ML(S) × ML(S) → ℝ and the stable lamination L has non zero intersection with every lamination which is not a multiple of itself.

By passing to subsequences, we may assume that x_{1}^{j}, x_{2}^{j}, x_{3}^{j}, … converges in P L(S) to some L_{j} where L_{m} = L. In ML(S) continuity forces i(L_{j}, L_{j+1}) = 0; j = 1, 2, …, m − 1 for any representatives of the indicated projective classes. By the previous paragraph and induction we see that L = L_{m} = L_{m−1} = ⋯ = L_{1}. This is impossible since L_{1} ∈ C. □

2.8 Remark. The above proof also establishes Theorem 2.5 with d in place of cd. I do not know the validity of Theorem 2.7 for cd.

3. Seifert manifolds. It is known [MS] that every splitting of an orientable Seifert manifold with orientable base space is either horizontal or vertical (definitions below). All Seifert manifolds have vertical splittings, but most do not admit horizontal splittings. Theorem 0.3 of [MS] describes, in terms of the Seifert invariants, those Seifert manifold which have horizontal splittings.

We show in this section that there is a strong restriction on vertical splittings of Seifert manifolds and a restriction on horizontal splittings as well. An easy corollary is that all splittings of closed, orientable Seifert manifolds are distance at most two splittings. The same is true for any closed, oriented 3-manifold which contains an essential torus.

3.1 Definition. For a splitting S = (S; V_{1}, V_{2}) we define the k-simplex intersection complexity of S to be:

c_{k}(S) = min{i(X_{1}, X_{2}) : X_{i} is a k-simplex in Ker(π_{1}(S) → π_{1}(V_{i})

without superfluous vertices}

Perhaps these give the most elementary measures of complexity for a splitting of a 3-manifold. It should be clear adding superfluous vertices to either of the X_{i} can only increase intersections (and dimension) without adding topological information, but it should not be assumed that in a fixed dimension the minimum i(X_{1}, X_{2}) occurs without superfluous vertices.

Now consider an orientable Seifert manifold M with base surface B and projection f : M → B. Suppose we have a cell decomposition of B such that B = D ∪ E ∪ F where each of D, E, F is a disjoint union of closed 2-cells of the decomposition, each component of D and of E contains at most one singular point, which is an interior point, each component of F is a square containing no singular point and having one pair of opposite sides in D and the other pair in E, Int(D) ∩ Int(E) = Int(D) ∩ Int(F) = Int(E) ∩ Int(F) = ∅, and D ∪ F and E ∪ F are connected. See Figure 3.

Then f^{−1}(D) is homeomorphic to D × S^{1}, the same holds for E and F, and we consistently fix such identifications. Let V_{1} = D × S^{1} ∪ F × [0, 1/2] and V_{2} = E × S^{1} ∪ F × [1/2, 1]; where S^{1} = [0, 1]/0 ∼ 1. Put S = V_{1} ∩ V_{2} = ∂(V_{1}) = ∂(V_{2}). Then (S; V_{1}, V_{2}) is a splitting of M of genus

g = β_{0}(D) + β_{1}(D ∪ F) = β_{0}(E) + β_{1}(E ∪ F) = 1 + β_{0}(F)

which we call a vertical splitting.

3.2 Theorem. Let S be an irreducible splitting of genus g ≥ 2 of a closed, orientable 3-manifold M. Then S is a vertical splitting of a Seifert manifold if and only if c_{g−2}(S) ≤ 2g − 2.

Proof. Given a vertical splitting of M as in the definition above, choose for each component F_{i} of F spanning arcs A_{i} and B_{i} of F_{i} meeting in a single point and joining the opposite edges of F_{i} which lie in D and E respectively. Then X_{1} = ∪∂(A_{i} × [0, 1/2]) and X_{2} = ∪∂(B_{i} × [1/2, 1]) are the desired g − 2 simplexes with i(X_{1}, X_{2}) = 2g − 2.

Now suppose we have a splitting (S; V_{1}, V_{2}) of M and for each i = 1, 2 a collection D_{i} = {D_{i,0}, D_{i,1}, …, D_{i,g−2}} of disjoint, properly embedded 2-cells in V_{i} such that no component of S − ∪_{j}∂(D_{i,j}) is planar and i(∂(D_{1}), ∂(D_{2})) ≤ 2g − 2.

Suppose S − ∪_{j}∂(D_{i,j}) has c_{i} components. By the reasoning of Lemma 1.1 the sum of their genera is c_{i}. Since no component is planar, they are all of genus one. Thus D_{i} splits V_{i} into a collection of solid tori.

If some D_{1,j} meets some D_{2,k} in a single point, the splitting has a trivial handle and is reducible contrary to assumption.

If some D_{1,j} does not meet D_{2} then ∂(D_{1,j}) lies in a solid torus component W of V_{2} split along D_{2} and we see that M contains a punctured Lens space ( possibly S^{3} or S^{2} × S^{1}). In the case of S^{3} or S^{2} × S^{1}, some meridian disk D of W meets ∂(D_{1,j}) in at most one point and can be chosen to have boundary in S. In the other cases M is either a Lens space or a non trivial connected sum. But Lens spaces have unique splittings [BO] and splittings of connected sums are reducible [H]; so in every case we get the contradiction that the splitting is reducible.

These observations, together with the assumption i(∂(D_{1}), ∂(D_{2})) ≤ 2g − 2 imply that each D_{1,j} must meet some component of D_{2}, which we assume is D_{2,j} in two points and be disjoint from every other component of D_{2}. Thus a regular neighborhood N of D_{1} ∪ D_{2} is the disjoint union of g − 1 solid tori. The j-th one can be identified with B^{2} ×S^{1} in such a way that D_{1,j} is identified with A_{1} ×[0, 1/2] and D_{2,j} with A_{2} × [1/2, 1] for some pair A_{1}, A_{2} of properly embedded arcs in B^{2} which cross once. Each component, say W, of Cl(V_{i} − N) is a solid torus which meets N in a collection of annuli in the boundary of each.

The product fibration of N will extend to a Seifert fibration of W unless one of these annuli is inessential in W. This, however, would give a punctured Lens space in M and a contradiction as before. Thus we can conclude that M is Seifert fibered and, in fact, see that the given splitting has the structure of a vertical splitting. □

3.3 Corollary. A vertical splitting of genus g of a closed Seifert manifold is a distance at most two splitting if g = 2 and a distance at most one splitting otherwise.

We define a horizontal splitting as follows. Take a surface bundle

N = F × [0, 1]/(x, 0) ∼ (ϕ(x), 1)

where F ≠ B^{2} is a compact, connected, orientable surface with one boundary component and ϕ : F → F is an orientation preserving homeomorphism with ϕ|∂(F) = 1. Then λ = ∂(F) × 0 and μ = x_{0} × [0, 1]/ ∼ form a basis for H_{1}(∂(N)). Let

M = N ∪_{h} B^{2} × S^{1}

be a Dehn filling of N where h : ∂(B^{2} × S^{1}) → ∂(N) is a homeomorphism such that h(∂(B^{2}) × 0) is homologous to μ + nλ for some n ∈ ℤ.

Now h^{−1}(∂(F) × {0, 1/2}) bounds an annulus A ⊂ B^{2} × S^{1} which splits B^{2} × S^{1} into two solid tori U_{1} and U_{2} with (U_{i}, A) homeomorphic to (I ×I ×S^{1}, I ×0×S^{1}). Then V_{1} = F × [0, 1/2] ∪_{h} U_{1} and V_{2} = F × [1/2, 1] ∪_{h} U_{2} are handlebodies which give a splitting of M which we call a horizontal splitting

If ϕ is (isotopic to) a periodic homeomorphism of F then N will be Seifert fibered and if n ≠ 0 this extends to a Seifert fibration of M (with a singular fiber in M −N when n ≠ ±1). However there is no need to assume this for the following.

3.4 Theorem. A horizontal splitting of a closed 3- manifold is a distance at most two splitting; in fact it has the form (S; V_{X}, V_{Y}) where every vertex of X is at distance at most two from some vertex of Y and vice versa.

Proof. Let (S; V_{1}, V_{2}) be a splitting obtained as in the above definition. There are homeomorphisms f_{i} : F × [0, 1] → V_{i} such that f = f_{2}^{−1} ∘ f_{1}|∂(F × [0, 1]) is level preserving, f(x, 0) = (x, 0), f(x, 1) = (ϕ(x), 1), and f(e^{iθ}, t) = (e^{inθ}, t) for (e^{iθ}, t) ∈ ∂(F) × [0, 1]).

Let A be a 1-manifold which splits F into a 2-cell. Then X = f_{1}(∂(A × [0, 1]) and Y = f_{2}(∂(A × [0, 1])) gives the desired diagram for the splitting. A component of A gives rise to a component of X and a component of Y which are at a distance of at most two as they don’t fill f_{1}(F × 1) = f_{2}(F × 0). □

3.5 Theorem. Let (S; V_{1}, V_{2}) be an irreducible, horizontal splitting of a closed Seifert manifold M. Then there are essential annuli A_{i} ⊂ V_{i}, i = 1, 2 with i(∂(A_{1}), ∂(A_{2})) ≤ 1

Proof. The base surface of M, homeomorphic to F/ϕ, is orientable. If it has positive genus or if M has more than three singular fibers, then M contains an essential torus which is a union of nonsingular fibers and which may be assumed to lie in N (as in the definition of horizontal splitting). This torus will meet V_{1} and V_{2} in essential annuli which may be moved slightly to be disjoint.

Lens spaces do not have irreducible splittings of genus g ≥ 2. So we are left with the case that M fibers over S^{2} with exactly three exceptional fibers.

The horizontal splitting comes from a Dehn filling on a surface bundle F × I/(x, 0) ∼ (ϕ(x), 1) for some periodic homeomorphism ϕ : F → F. The orbit map f : M → S^{2} induces a cyclic branched covering f : F → B^{2} with ϕ generating the group ( ≅ ℤ_{n}) of covering transformations, and this, in turn, is determined by an epimorphism π_{1}(B^{2} − {branch points}) → ℤ_{n}.

Now F is obtained from an annulus A by identifying edges, in pairs, on one component of ∂(A) and ϕ is induced by an equivariant rotation of A. To see this choose an arc a ⊂ Int(B^{2}) which contains the branch points. Then π_{1}(B^{2}−a) → ℤ_{n} is an epimorphism (as ∂(F) is connected); so f^{−1}(B^{2} − a) is a half open annulus in F which we can complete to get the desired A.

Some arc in A whose end points are midpoints of identified edges gives rise to an essential simple closed curve J in F. Otherwise we get the contradiction that F = B^{2}. Moreover we can adjust so that #(J ∩ ϕ(J)) = i(J, ϕ(J)) ≤ 1. Then A_{1} = J × [0, 1/2] and A_{2} = J × [1/2, 1] (adjusted to general position) give the desired annuli. □

3.6 Lemma. Let (S; V_{1}, V_{2}) be a strongly irreducible splitting of a closed 3-manifold M which contains an essential torus or Klein bottle T. Then, after an isotopy of T we may assume that each component of T ∩ V_{i}, i = 1, 2 is an essential annulus or Möbius band in V_{i}.

Proof. . The sum of the Euler characteristics of the components of T split along S ∩ T is χ(T) = 0. So if there are no disk components, then all components are annuli or Möbius bands which are incompressible in the V_{i} in which they lie. Some of these annuli might be parallel to annuli in S and could be eliminated by an isotopy of T, but some must remain; as T cannot be isotoped into a handlebody.

So we induct on the number of disk components of T split along S ∩T. Since the splitting is assumed to be strongly irreducible, there are not disk components in both V_{1} and V_{2}. We assume they are all in V_{1}. Choose a component C of T ∩V_{2} such that ∂(C) contains a simple closed curve J which bounds a disk D ⊂ T ∩V_{1}. J does not bound a disk in V_{2}; so it must meet some meridian disks for V_{2}. This means that there is a boundary compression of C along an arc a ⊂ C which cobounds a disk in V_{2} − T with an arc b ⊂ S where a has an endpoint in J. The boundary compression eliminates D as a component of T ∩ V_{1}: replacing it by an annulus, if both endpoints of a are in J or by the connected sum of D with another component of T ∩ V_{1} otherwise.

If C were an annulus it would be reduced to a disk in V_{2} by the boundary compression. This, however, would contradict strong irreducibility. There is no other way in which new disk components could be introduced; so induction applies to complete the proof. □

3.7 Corollary. Any strongly irreducible splitting of a closed 3-manifold M which contains an essential torus or Klein bottle is a distance two splitting.

Proof. Let (S; V_{1}, V_{2}) be a strongly irreducible splitting of our manifold M. By 3.6 there are essential annuli or Möbius bands A_{i} ⊂ V_{i}, i = 1, 2 which have a common boundary component J. A boundary compression of A_{i} gives rise to a disk D_{i} ⊂ V_{i} which, since A_{i} is not parallel to an annulus in S, is essential in V_{i}. We may assume ∂(D_{i}) ∩ ∂(A_{i}) = ∅. Thus d(∂(D_{i}), J) ≤ 1 and the proof is complete. □

3.8 Corollary. Any splitting of a closed, orientable Seifert manifold is a distance at most two splitting.

Proof. For Seifert manifolds with orientable base this follows from 3.3, 3.4 and Theorem 0.1 of [MS]. Seifert manifolds with nonorintable base must contain an essential torus or Klein bottle and 3.7 applies. □

4. Complexity bounds. Casson and Gordon [CG2], [K] gave a rectangle condition on a Heegaard diagram which implies that it determines a strongly irreducible splitting. We give here a quantitative version of this condition which gives lower bounds for the complexity c_{k}(S), and in particular is used, in Theorem 4.4, to show that a diagram does not determine a vertical splitting of a Seifert manifold manifold. We also give a quantitative version of the strong rectangle condition introduced by Kobayashi [K] to give conditions that the splitting determines an atoridal manifold. The quantitative version is used in corollary 4.7 to give criteria that a splitting not be a horizontal splitting of a Seifert manifold.

Somewhat stronger versions of the results are available for genus two splittings and are presented separately.

A pair X, Y of simplexes of C(S) determines a cell structure on S whose faces are the components of S−(X ∪Y) (assuming these are all simply connected). Every vertex has order four and every face has an even number (≥ 4) of edges which lie alternately in X and Y. A standard calculation gives.

4.1 Lemma. If X and Y are simplexes of C(S) with S −(X ∪Y) simply connected and having n_{i} 2i-gon components (i = 1, 2, …), then

χ(S) = ∑(1 − i/2)n_{i}

Since n_{1} = 0 (assuming efficient intersection) and χ(S) < 0, most of the complementary regions will be squares with one pair of opposite edges in X and the other pair in Y. If we “stack” together adjacent squares along common edges in X maximally we get an X-stack. The top and bottom edges will lie in large ( ≥ 6 sides) regions, and the sides will lie in (possibly the same) component(s) of Y. The process must actually stop at a top and bottom; otherwise we would have two parallel components of Y. The Y -stacks are defined by interchanging the roles of X and Y. The number of squares in a stack is called its height. For logical consistancy we must include stacks of height 0 – corresponding to edges common to two large complementary regions. These occur rarely and will never satisfy the conditions of our theorems.

For a somewhat different picture, split S open along Y. X gets split into a collection of arcs which fall into familys of parallel arcs which correspond to the X-stacks; where a family with h + 1 arcs corresponds to a stack of height h. A component of Y will contain sides of some X-stacks which lie to either of its sides; an X-stack on one side may meet several X-stacks from the other side. The union of the X-stacks is a regular neighborhood of a train track which has one branch for each X-stack. This train track carries X with weights the numbers: stack height +1.

If X and Y are full simplexes forming a generic pair. Then each X-stack lies in a component P of S split along Y, and P is planar. The sides of the stack must lie in different boundary components of P (but which could be identified to the same, essential, component of Y); otherwise there would be a wave. In fact every potential wave (arc in P with ends in the same boundary component of P which is not parallel to an arc in this boundary component) must cross some X-stack (possibly of 0 height). This gives part of:

4.2 Lemma. If X, Y is a generic pair of full simplexes of C(S) then there are the same number of X-stacks as Y -stacks. This number is ∑ n_{i}/2 and it lies in the interval [2max{dim(X), dim(Y)} + 2, 6g(S) − 6].

Proof. The function that assigns to each X-stack its bottom (with respect to some arbitrary orientation) gives a bijection between the set of X-stacks and half of the edges of large regions. This is clearly symmetric in X and Y.

When we split S along Y we get dim(Y) + 2 − g component, each planar, with a total of 2dim(Y) + 2 boundary components . It takes at least p stacks to block a wave in a planar region with p boundary components.This gives the lower bound

If we collapse the boundary components of S split along Y to points and collapse the X-stacks to arcs, we get a cell structure on the disjoint union of dim(Y) + 2 − g 2-spheres (g = g(S)) whose edges correspond to the X-stacks and whose number is at most the number of edges in a triangulation with the same number of vertices, which is 6g − 6. □

The intersection number of an X-stack and a Y -stack is the number of squares common to the two stacks. The stack intersection matrix for a pair X, Y is the matrix of intersection numbers of non empty (but possibly height 0) X-stacks and Y -stacks. A stack of height 0 will give a row or column of 0’s to this matrix. The matrix will have ∑ n_{i}/2 rows and columns. If all the large complementary regions are hexagons, this number will be 6g − 6. This holds, in particular, when X and Y are maximal dimension (= 3g − 4) simplexes forming a generic pair. Then each component of S split along Y contains three X-stacks: one with sides in each pair of boundary components (and vice versa).

The stack intersection matrix defines an integer valued bilinear form which we can use to estimate intersection numbers as follows. Suppose A is a simplex of C(S) which has been isotoped so as to meet X and Y efficiently. If t_{j} is an Y - stack, a component of A ∩ t_{j} which does not meet the sides of t_{j} must have one end point in the top and one in the bottom of t_{j}. We call such a component a stack crossing and denote the number of such by a_{j} ≥ 0. So we get a stack crossing vector (a_{1}, a_{2}, …) of A with respect to Y. If another simplex B of C(S) has stack crossing vector (b_{1}, b_{2}, …) with respect to X then in each square common to Xstack s_{i} and Y -stack t_{j} we see b_{i}a_{j} points of A ∩ B. This gives

4.3 Lemma. Let X, Y be a generic pair of simplexes of C(S) with corresponding stack intersection matrix (u_{i,j}). Suppose A and B are simplexes of C(S) which meet efficiently and which have stack crossing vectors (a_{1}, a_{2}, …) with respect to Y and (b_{1}, b_{2}, …) with respect to X. Then

i(A, B) ≥ ∑_{i,j} b_{i}u_{i,j}a_{j}

4.4 Theorem. Let X, Y be a generic pair of simplexes of C(S) with corresponding stack intersection matrix (u_{i,j}). Then c_{0}(S; V_{X}, V_{Y}) ≥ 4min{u_{i,j}}; so if u_{i,j} > 0 for all i, j the splitting (S; V_{X}, V_{Y}) is strongly irreducible and is not a vertical splitting of a Seifert manifold.

Proof. Let a and b be vertices of K_{X} and K_{Y} respectively. We may assume that all pairs of curves from X, Y, a, b meet efficiently. Suppose that a ∩ X ≠ ∅. Look at how a disk in V_{X} bounded by a meets disks bounded by the components of X. There must be at least two “outermost” arcs of intersection. The arcs on a with the same boundary must contain Y -stack crossings – otherwise there would be a wave. If a ∩ X = ∅, then a ∩ Y ≠ ∅ – otherwise S − X ∪ Y is not simply connected. Thus a meets some Y -stack in a stack crossing. If there were only one Y - stack crossing on a we could replace a neighborhood of this crossing on a by a pair of arcs running parallel to the top and bottom of the stack and ending in a component of X. There are two ways of doing this. One will produce a wave.

Thus a contains two Y -stack crossings (possibly of the same stack. Similarily b contains two X-stack crossings. It follows from Lemma 4.3 that i(a, b) ≥ 4min{u_{i,j}}. The final conclusion then follows from Theorem 3.2 □

This theorem can be improved for genus two splittings due to the fact that a and b must then contain crossings of two distinct stacks (though not necessarily associated with outermost arcs). So let g(S) = 2 and let X = (x_{0}, x_{1}, x_{2}) and Y = (y_{0}, y_{1}, y_{2}) be generic 2-simplexes in C(S) without essential vertices.

Each x_{i} (y_{j}) misses two Y -stacks (X-stacks): those with sides in the other two components of X (Y). Let c_{i,j} be the sum of the corresponding four stack intersection numbers.

4.5 Theorem. Let X and Y be a generic pair of 2-simplexes in C(S) without essential vertices; where g(S) = 2. Then c_{0}(S; V_{X}, V_{Y}) ≥ min{c_{i,j}}.

Proof. Choose vertices a ∈ K_{X}, b ∈ K_{Y} as before. Consider first the case a ∩ X ≠ ∅ ≠ b ∩ Y. Now a meets each of the (pants) components P_{1}, P_{2} of S split along X in at most three families of parallel arcs. Corresponding to an outermost arc, one of these, say P_{1}, contains a family of n ≥ 1 parallel arcs in a whose end points lie in the same component, say x_{i} of X. Consider the various possibilities for the families of arcs in a ∩ P_{2} and note that there are matching equations equating the number of endpoints on each component of X comming from the two sides. There is only one solution (this is the basis of Neilsen-Fenchel coordinates, c.f [P2]): a must meet P_{2} in a family of n parallel arcs with both end points in x_{i}.

Similarly we see that for some j and some integer m ≥ 1 that b must meet each component of S split along Y in families of m parallel arcs with both end points in y_{j}. It then follows from Lemma 4.3 that i(a, b) ≥ nmc_{i,j}.

Now consider the case a ∩ X = ∅ but with b as above. Then a is isotopic to a component of X, say x_{0}. We can isotope a to either side of x_{0} and each time get a lower bound on i(a, b) from Lemma 4.3. Averageing these gives i(a, b) ≥ m(c_{1,j} + c_{2,j})/2.

Simerlarly one can show, for example, that i(x_{0}, y_{1}) ≥ (c_{1,0} + c_{1,2} + c_{2,0} + c_{2,2})/4 to handle the case a ∩ X = b ∩ Y = ∅. □

4.6 Theorem. Let X, Y be a generic pair of (3g−4)-simplexes in C(S) (g = g(S)) such that the 3g − 5 faces X′ of X and Y′ of Y are all full and each pair X′, Y′ of them is generic, has all large complementary regions hexagons, and has all stack intersection numbers at least u. If A is an essential annulus in V_{X} and B is an essential annulus in V_{Y}, then i(∂(A), ∂(B) ≥ u.

Proof. The goal is to find 3g − 5 faces X′ of X, Y′ of Y such that ∂(A) contains an Y′-stack crossing and ∂(B) contains a X′-stack crossing and then apply Lemma 4.3. A crossing of a Y -stack by ∂(A) (or a Y -stack by ∂(B) will give one for every codimension one face ; so we assume none exist.

The intersection of A with the union D of the disks bounded by the components of X will be a collection of arcs splitting A into 2-cells. The graph in A dual to these arcs is a deformation retract of A and thus has Euler characteristic zero. If this graph has a vertex of order different than 2, it must have a vertex of order 1. This vertex comes from an outermost arc and indicates the presence of an Y -stack crossing in ∂(A).

So we assume all the vertices of the dual graph have order 2 and thus the components of A split along D will all be squares with a pair of opposite sides in D and the other pair in S. The first pair must lie in different components of D; otherwise there would be an Y -stack crossing. Each such square lies in some component C of V_{X} split along D. C is a 3-cell which meets D in three 2-cells in its boundary. The square misses exactly one of these 2-cells and this gives a preferred direction in which to isotope the square into S – away from the 2-cell it misses. If these directions agree from square to square, we get the contradiction that A is parallel to an annulus in S.

Thus there must be adjacent squares s_{1}, s_{2} in adjacent components C_{1}, C_{2} of V_{X} split along D such that ∂(s_{1} ∪ s_{2}) separates some pair x_{i}, x_{j} of components of X on ∂(C_{1} ∪ C_{2}) (see Figure 4).

We get the desired face X′ by deleting the component of X lying in C_{1} ∩C_{2}. For any codimension one face Y′ of Y there is, by assumption, a Y′-stack with sides in x_{i} and x_{j} which therefore must be crossed by ∂(A). □

4.7 Corollary. With the hypothesis and notation of Theorem 4.6, if u > 0 then M_{X,Y} contains no essential torus. If u > 1 then (S; V_{X}, V_{Y}) is not a horizontal splitting of a Seifert manifold.

Proof. This follows immediately using 3.5 and 3.6 and the observation that the splitting is strongly irreducible by 4.4. □

4.8 Comments.

1. Suppose (S : X′, Y) is obtained from (S; X, Y) by a generic expansion. The square regions of S − X′ ∪ Y will be “halves” of squares of S − X ∪ Y split by the new curve together with some new squares cut off of large regions of S − X ∪ Y by the new curve. A new square could form a X′-stack of height one (if two large regions of S − X ∪ Y share an edge), but in general they get added to (subdivided) X-stacks and one can show that the stack intersection numbers go up. For this reason the theorems of this section generally give the best results when applied to principal simplexes X, Y – that is pants decompositions.

2. Theorem 4.5 is definitely stronger than Theorem 4.4 applied to genus two splittings. Consider the example of Figure 10(a). Even when extended to pants decompositions, some stack intersection numbers will be zero, but the minimal c_{i,j} will be 2.

5. Distance three splittings. In this section we give a criterion for recognizing distance three splittings and apply it to give some examples.

For simplexes X and Y of C(S), s an X-stack and t a Y -stack, s ∪ t and Cl(S−s∪t) will be 2-manifolds except possibly at a finite number of singular points where a corner of s meets a corner of t. A regular neighborhood of Cl(S − s ∪ t) will be called a complementary region and denoted CR(s ∪ t). One gets one such by adding to Cl(S − s ∪ t) a suitable neighborhood of the singular points. We say that s ∪ t fills (almost fills) S if the components of CR(s ∪ t) are all 2-cells ( 2-cells and annuli)

5.1 Theorem. Let X, Y be a generic pair of full simplexes of C(S) without essential vertices such that for every X-stack s and Y -stack t s ∪ t fills S. Then (S; V_{X}, V_{Y}) is a distance ≥ 3 splitting.

Proof. Suppose there are vertices a ∈ K_{X}, b ∈ K_{Y} with d(a, b) ≤ 2. So there is some vertex c ∈ C(S) such that c ⊂ S −a∪b. Now a must cross some Y -stack t and b must cross some X-stack s. Then c misses the stack crossings and therefore must be isotopic into CR(s ∪ t). This fact requires the assumption that X and Y have no essential vertices; so that the stacks are embedded rectangles. In particular, if two squares of s ∪ t intersect in an edge lying in Y then at most one of the squares can lie in X and vice versa. It follows from the observation that the regions of s ∪ t complementary to the crossing arcs either lie in the interior of one of the stacks or is a rectangle or annulus (it has a cell decomposition by rectangles) meeting ∂(s∪t) in a connected set. It of course contradicts the assumption that s ∪ t fills S. □

5.2 Theorem. Let X, Y be a generic pair of full simplexes of C(S) without essential vertices such that for every X-stack s and Y -stack t s ∪ t almost fills S. Then either (S; V_{X}, V_{Y}) is a distance ≥ 3 splitting or for some X-stack s and Y -stack t and some component R of CR(s ∪ t), S − R is compressible in both V_{X} and V_{Y}.

Proof. We proceed just as with 5.1: If the splitting is not a distance ≥ 3 splitting we find a vertex C ∈ C(S) disjoint from vertices a ∈ K_{X}, b ∈ K_{Y} and lying in some component R of CR(s ∪ t) for some X-stack s and Y -stack t. If R is an annulus, then a and b are isotopic into S − R; so S − R is compressible in both V_{X} and V_{Y}. □

Note : there is an algorithm for deciding whether a surface in the boundary of a handlebody is compressible in the handlebody. It seems fairly well known, but we state it for completeness. It comes from repeated applications of the following.

5.3 Lemma. Let X be a full simplex of C(S) and F be a compact surface in ∂(V_{X}) whose boundary meets X efficiently. Suppose that F split open along X is simply connected. Then:

(1) If F is compressible in V_{X} then there is a wave of X lying in F, and

(2) If there is a wave of X in F, then surgery of X along this wave produces a simplex X′ of C(S) with V_{X′} = V_{X} and i(X′, ∂(F)) ≤ i(X, ∂(F)) − 2.

By a Dehn twist along a simplex X ⊂ C(S) we mean the product of the (commuting) Dehn twists along the vertex curves of X.

5.4 Theorem. Let X, Y be simplexes of C(S) with Y full and having no essential vertices. Suppose that for each X-stack s, S − s ∪ X is simply connected. Let h : S → S be the Dehn twist along X. Then for n ≥ 2 (S; V_{h^{n}(Y)}, V_{Y}) is a distance ≥ 3 splitting.

Proof. We will use Theorem 5.1. The following describes h^{n}(Y). For each component x_{i} of X let k_{i} = i(x_{i}, Y) and let A_{i} be an annular neighborhood of x_{i}. Replace each of the k_{i} arcs of A_{i} ∩Y by arcs which circle A_{i} n times (and smooth to general position relative to Y) to get a collection of curves representing h^{n}(Y). See Figure 5.

The regions of S − h^{n}(Y) ∪ Y are of two types. There are the old regions which are essentially the regions of S − X ∪ Y, but whose boundary has been twisted – with some edges shrunk to near (old) vertices and alternate edges expanded to near two (old) edges. Then there are the new squares. They come in partial Y -stacks each of which begins at an old region on one side of some A_{i}, circles A_{i} (nk_{i} −1)/k_{i}. times and ends at an old region on the other side of A_{i}. There are k_{i} of these partial Y -stacks in each A_{i}.

The old regions at one or both ends of a partial Y -stack of (new) squares may be squares; so that the (new) Y -stacks, relative (h^{n}(Y), Y), will consist of these partial stacks joined together along old squares. The top and bottom will lie in old large regions. In particular, every old square in an X-stack, relative to (X, Y) will lie in a fixed new Y -stack. The hypothesis necessitates that each of these X-stacks meets each component of X. So each new Y -stack contains partial stacks circling each A_{i}.

The h^{n}(Y)-stacks, relative to (h^{n}(Y), Y) occupy essentially the same space as the X-stacks, relative to (X, Y); in fact we may assume that each of the latter lies in a unique one of the former.

So for each h^{n}(Y)-stack s and each Y -stack t , relative to (h^{n}(Y), Y) there is an X-stack s′, relative to (X, Y) and a subset of s ∪ t which misses any of its singular vertices and is isotopic to s′ ∪ X. Here we use the assumption n ≥ 2 to show that each partial stack crosses s′ twice and so s′∪ partial stack contains, up to isotopy, s′ ∪ x_{i}.

The conclusion now follows from Theorem 5.1 □ 5.5 Example. The following diagram, Figure 6, satisfies the hypothesis of Theorem 5.4 and so provides examples of distance ≥ 3 splittings.

6. Distance two, genus two splittings. In this section we describe all 3-manifolds which admit such splittings. We show that there is always a torus separating the manifold into pieces of two specific types. Either type may reduce to a solid torus in special cases (which we describe). The details of this decomposition can be read off from a particularly nice diagram of the manifold.

One of the pieces will be Seifert fibered over B^{2} with at most two singular fibers. We call such a manifold a generalized torus knot space or simply a GTS and denote it by GT S(β_{1}/α_{1}, β_{2}/α_{2}) to indicate the fiber invariants. It is the complement of an open regular neighborhood of a “torus knot” (i.e. lying on a splitting torus) in a lens space. Completing GT S(β_{1}/α_{1}, β_{2}/α_{2}) to a Seifert fibration over S^{2} with Euler number b produces the lens space

L_{α_{1}α_{2}b−α_{1}β_{2}−α_{2}β_{1},q}.

The following is immediate from the classification of Seifert manifolds.

6.1 Lemma. GT S(β_{1}/α_{1}, β_{2}/α_{2}) is a solid torus if and only if |α_{1}| = 1 or |α_{2}| = 1.

The other type of piece might be called a one-bridge in a lens space knot complement as it is the complement of a neighborhood of a knot which lies, except for one bridge, on a splitting torus for a lens space. By this reasoning we perhaps should be calling a GTS a zero-bridge in a ..., but we won’t do either. We use the expression OBL to refer to such a manifold.

An OBL has the following structure. Let R = T − Int(D_{1} ∪ D_{2}) where T is a torus and D_{1} and D_{2} are disjoint disks in T. Let a_{0} and a_{1} be simple closed curves in R which meet efficiently. Then

R × [0, 1] ∪_{a_{0}×0} 2 − handle ∪_{a_{1}×1} 2 − handle

is an OBL which we denote by OBL(a_{0}, a_{1}) and every OBL is so obtained.

Note that (T; a_{0}, a_{1}) is a diagram for a lens space L. Let k_{i} be an arc from the center of D_{1} to the center of D_{2} in T − a_{i} and let k′_{0} be an arc in L obtained by pushing Int(k_{0}) into V_{0}. Then k = k′_{0} ∪ k_{1} is a knot in L and the complement of a neighborhood of k is homeomorphic to OBL(a_{0}, a_{1}).

Let < , >: H_{1}(S) × H_{1}(S) → ℤ denote the intersection pairing on a surface S. So for oriented simple closed curves x, y in S meeting efficiently < x, y >= i(x, y) means that the intersection number is +1 at each point of x ∩ y.

6.2 Lemma. OBL(a_{0}, a_{1}) is a solid torus if and only if | < x, y > | = i(x, y) and either i(a_{0}, a_{1}) = 1 or D_{1} and D_{2} lie in regions of T − a_{0} ∪ a_{1} having an edge in common.

Proof. First we construct a genus two diagram for M = OBL(a_{0}, a_{1}). We may assume that D_{1} and D_{2} are (small) disks centered at points of a parallel copy a′_{0} of a_{0} and that the attached handle misses a′_{0} × 0. Then a′_{0} − Int(D_{1} ∪ D_{2} will be the union of two arcs b_{1} and b_{2} such the disks E_{i} = b_{i} × [0, 1], i = 1, 2 split V = R × [0, 1] ∪_{a_{0}×0} 2 − handle into a 3-cell. Thus V is a handlebody and (∂(V); {x_{1}, x_{2}}, y) is a diagram for M where x_{i} = ∂(E_{i}) and y = a_{1} × 1. See Figure 7.

Since some covering space of M embeds in S^{3}, M is irreducible. Thus M is a solid torus if and only if ∂(M) is compressible in M . This in turn holds if and only if π_{1}(M) = ℤ. We break the proof into two cases.

First suppose that | < a_{0}, a_{1} > | < i(a_{0}, a_{1}). We will show in this case that there is no wave of X in ∂(V) − y; so by 5.3 ∂(V) − y is incompressible in V. It follows from a theorem of Jaco [J] (valid for a single 2-handle attachment) that ∂(M) is incompressible in M to complete the proof in this case.

Now geometric and algebraic intersection numbers agree for curves meeting efficiently on a torus. Thus there are exactly two bigon regions of T − a_{0} ∪ a_{1} and they contain D_{1} and D_{2} respectively. They lie on opposite sides of both a_{0} and a_{1}. The situation must be as shown in Figure 8.

Now A = ∂(V) − R × 1 is an annulus. It meets each of x_{1}, x_{2} in a spanning arc of A. The two (square) components of A − x_{1} ∪ x_{2} might serve as paths for a wave to move from one part of R × 1 to another. But they don’t: each sees a “dead end” associated with one of the bigons. That is a potential wave could be isotoped to lie in R. But from there it can be seen not to exist at all.

Now assume a_{0} and a_{1} meet positively at every point. In this case there may be waves – even when ∂(V) − y is incompressible; so we need a different argument.

Let r = i(a_{0}, a_{1}). If r = 0, then π_{1}(M) = ℤ ∗ ℤ; so M is not a solid torus. If r = 1 then M is a solid torus. Thus we assume r ≥ 2. If D_{1} and D_{2} lie in the same component of T − a_{0} ∪ a_{1}, then π_{1}(M) = ℤ ∗ ℤ_{r}; so we assume this does not happen.

Now we may assume that T = ℝ^{2}/ℤ^{2}, a_{0} is the image of the y-axis, a_{1} is the image of y = sx/r (where (r, s) = 1), and D_{i} is a small disk centered just to the right of (0, 1/2r) for i = 1 and (0, (t + 1/2)r) for i = 2. See Figure 9. We refer to M as OBL^{+}(s/r, t)

Corresponding to the splitting described above we have the presentation:

π_{1}(OBL^{+}(s/r, t)) =< x_{1}, x_{2} : x_{i_{1}}x_{i_{2}} … x_{i_{r}} = 1 >

where i_{j} = 1 if [1 + (j − 1)s] ∈ {1, 2, …, t} and i_{j} = 2 otherwise. Here [n] denotes the least non-negative residue of n modulo r. We will complete the proof in this case by showing that π_{1}(M) ≅ ℤ if and only if t ≡ ±1, ±s mod r.

If t = ±1 then the relation involves just one occurrence of x_{1} or x_{2}, and clearly π_{1}(M) = ℤ. The case t = ±s reduces to the case t = ±1 on interchanging the roles of a_{0} and a_{1}.

By obvious symmetries (which may reverse orientation) we may assume that 1 < t ≤ r/2 and 1 ≤ s < r/2. By combining terms, and conjugating, if necessary, we can express the relation in the form

x_{1}^{n_{1}}x_{2}^{m_{1}}x_{1}^{n_{2}}x_{2}^{m_{2}} … x_{1}^{n_{k}}x_{2}^{m_{k}}

where n_{i}, m_{i} ≥ 1. The n_{i} (respectively m_{j}) are the lengths of maximal sequences [q], [q + s], [q + 2s], … lying in {1, 2, …, t} (respectively {t + 1, t + 2, …, r}).

If s < t (and so s < r − t) then some n_{i} ≥ 2 and some m_{j} ≥ 2. If s > t, then, say, every n_{i} = 1, but using s < r/2 we can show that some two m_{j}’s differ by at least two. In either case Lemma 6.3 below applies to show π_{1}(M) ≅ ℤ̸. □

6.3 Lemma. Let G =< x_{1}, x_{2} : x_{1}^{n_{1}}x_{2}^{m_{1}}x_{1}^{n_{2}}x_{2}^{m_{2}} … x_{1}^{n_{k}}x_{2}^{m_{k}} = 1 > be a group presentation with n_{i} > 0, m_{j} > 0 for all i, j. If either some n_{i} ≥ 2 and some m_{j} ≥ 2 or, say, some two m_{j}’s differ by at least 2 , then the Alexander polynomial Δ_{G}(z) is not constant. So G ≅ ℤ̸ ∗ H for any group H.

Proof. Let n = ∑ n_{i}, m = ∑ m_{i}, d = (n, m), n = dn̄, and m = dm̄. Then G/G′ = ℤ × ℤ_{d} and we have an epimorphism ϕ : G →< z : > of G to the (multiplicative) infinite cyclic group given by ϕ(x_{1}) = zm̄, ϕ(x_{2}) = z^{−n̄} which is unique up to the automorphism z → z^{−1}. We use the same notation for the extension ℤG → ℤ[z, z^{−1}].

The Alexander polynomial, Δ_{G}(z), of G is a generator for the ideal

(ϕ(∂r/∂x_{1}), ϕ(∂r/∂x_{2}).

Now (∂r/∂x_{1})(x_{1} − 1) = (∂r/∂x_{2})(1 − x_{2}). Since (n̄, m̄) = 1, the greatest common divisor of

ϕ(x_{1} − 1) = zm̄ − 1 and ϕ(1 − x_{2}) = 1 − z^{−n̄} = z^{−n̄}(zn̄ − 1)

is z − 1. It follows that

Δ_{G}(z) = ϕ(∂r/∂x_{1})/(1 + z + ⋯ + z^{n̄−1}) = ϕ(∂r/∂x_{2})/(1 + z + ⋯ + z^{m̄−1}).

Now for the first part of the theorem we may assume that n̄ ≥ m̄ and that some m_{j} ≥ 2. After cyclically permuting, if necessary, we may further assume r = wx_{1}x_{2}^{m_{j}}x_{1}u for some positive words w, u in the generators.

Then ∂r/∂x_{1} contains the terms w and wx_{1}x_{2}^{m_{j}} which, since all exponents are positive, are not cancelled by any other terms. The difference of degrees of the images of these terms under ϕ is |m̄ − n̄m_{j}| ≥ n̄. Thus we are left with a nonconstant (Laurant) polynomial after dividing by 1 + z + ⋯ + z^{n̄−1}.

For the second part suppose n_{1} = n_{2} = ⋯ = 1, that m_{1} is the smallest m_{j}, and that m_{j} −m_{1} ≥ 2 for some j. We replace x_{1} by x = x_{1}x_{2}^{m_{1}} to get a new presentation G =< x, x_{2} : x^{2}x_{2}^{m_{2}−m_{1}} … xx_{2}^{m_{j}−m_{1}} ⋯ > to which the first part applies.

For the final part suppose G ≅ ℤ ∗ H for some group H. Then H/H′ ≅ ℤ_{d}. So elements of H map to 1 under ϕ : G →< z : >. We get another presentation for G by adding a generator, but no relations, to a presentation for H. The entries of the Jacobian matrix of this presentation map to integers under ϕ. Thus Δ_{G}(z) is constant. □

6.4 Theorem. Let (S; V_{1}, V_{2}) be a genus two, distance two splitting of a 3-manifold M. Then there is a torus T ⊂ M splitting M into a GTS, M_{1}, and an OBL, M_{2}. There is a diagram (S; {x_{1}, x_{2}}, {y_{1}, y_{2}}) for the splitting so that S−x_{1}∪y_{1} contains an essential, nonseparating simple closed curve z, and for any such diagram

M_{1} = GT S(β_{1}/ < z, x_{2} >, β_{2}/ < z, y_{2} >)

and

M_{2} = OBL(S − z; x_{1}, y_{1})

Proof. Let (S; X, Y) be any diagram for the splitting. By assumption there are vertices x ∈ K_{X}, y ∈ K_{Y} with d(x, y) = 1. So there is an essential simple closed curve z ⊂ S − x ∪ y. We may assume z does not separate S; for otherwise x ∪ y, being connected, would lie in one component of S − z and we could replace z by an essential, nonseparating simple closed curve in the other component. Similarly we may assume x and y are nonseparating. For if, say, x separates S then each component of S − x contains a nonseparating vertex of K_{X} and we could replace x by one missing z. Thus we can extend x, y to a diagram as desired.

Now given any such diagram take disks D_{i} ⊂ V_{1}, E_{j} ⊂ V_{2} bounded respectively by x_{i}, y_{j}. Let A be a regular neighborhood of z in S, and let R = Cl(S − A) – a twice punctured torus. A regular neighborhood M_{2} of R ∪ D_{1} ∪ E_{1} in M is an OBL(x_{1}, y_{1}).

M_{1} = Cl(M − M_{2}) is the union of two solid tori: V_{1} split along D_{1} and V_{2} split along E_{1} pushed slightly away from R. These solid tori meet along A which circles them < z, x_{2} > and < z, y_{2} > times respectively.

Of course, the β_{i}’s can also be read off from the diagram as can the gluing map ∂(M_{1}) → ∂(M_{2}). Perhaps it is better not to try to squeeze too much in the statement of the theorem, but to leave it to calculation as illustrated in the examples below. □

6.5 Comments.

1. An OBL may be Seifert fibered. The “one bridge” might be isotoped back onto the torus T. One condition for being able to do this for OBL^{+}(s/r, t) is: one of A_{i} ∩ B_{j} = ∅ where

A_{1} = {1, 2, …, t − 1}, A_{2} = {t + 1, t + 2, …, r − 1}

B_{1} = {s, 2s, …, (s̄ − 1)s}, B_{2} = {(s̄ + 1)s, (s̄ + 2)s, …, (r − 1)s}; s̄s ≡ t mod r.

In particular every OBL^{+}(1/r, t) is Seifert fibered, as is, for example OBL^{+}(3/10, 6).

2. The complement (in S^{3}) of a 2-bridge knot is OBL(a_{0}, a_{1}) where there is a simple closed curve a′_{1} isotopic to a_{1} in T such that i(a_{0}, a′_{1}) = 1 and a′_{1} meets the projection of the knot in T in a single point.

3. There are OBL’s with Δ(0) ≠ ±1 and which therefore cannot be surface bundles over S^{1}; for example OBL^{+}(3/14, 7).

6.6 Examples. Each of the diagrams of Figure 10 satisfies the conditions of Theorem 6.4.

In (a) there is a torus splitting the manifold into a GT S(1/2, 1/3) and an OBL^{+}(1/5, 3) which by 6.5(1) is also Seifert fibered. This is a non-trivial graph manifold.

In (b) we have a splitting into a GT S(1/3, 1/4) and an OBL^{+}(1/3, 1) which by 6.2 is a solid torus. Thus the manifold is Seifert fibered. This could also be seen from Theorem 3.2 as the natural way of completing the diagram to pants decompositions will give i(x_{3}, y_{3}) = 2.

In (c) we have a splitting into a GT S(1/1, 1/3), which is a solid torus, and an OBL(x_{1}, y_{1}), which by 6.5(2) is a 2-bridge knot complement. With a little more effort this can be seen to be 2/3 surgery on the figure eight knot.

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Mathematics Department Rice University, Houston, TX 77251 Current address: MSRI, 1000 Centennial Drive, Berkeley, CA 94720

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