ON THE GEOMETRIC AND THE ALGEBRAIC RANK OF GRAPH MANIFOLDS

JENNIFER SCHULTENS AND RICHARD WEIDMANN

Abstract. For any n ∈ ℕ we construct graph manifolds of genus 4n that have 3n generated fundamental group.

1. introduction

A Heegaard surface of an orientable closed 3-manifold M is an embedded orientable surface S such that \overline{M − S} consists of 2 handlebodies V_{1} and V_{2}. This decomposition of M is called a Heegaard splitting and denoted by M = V_{1} ∪_{S} V_{2}. We say that the splitting is of genus g if S is of genus g. It is not difficult to see that any orientable closed 3-manifold admits a Heegaard splitting. If M admits a Heegaard splitting of genus g but no Heegaard splitting of smaller genus then we say that M has Heegaard genus g and write g(M) = g.

Clearly any curve in a handlebody can be homotoped to its boundary. It follows that for any Heegaard splitting M = V_{1} ∪_{S} V_{2} every curve in M can be homotoped into V_{1}. Thus the map induced by the inclusion of V_{1} into M maps a generating set of π_{1}(V_{1}) to a generating set of π_{1}(M). As π_{1}(V_{1}) is generated by g elements it follows that π_{1}(M) is also generated by g elements. Thus g(M) ≥ r(M) where r(M) denotes the minimal number of generators of π_{1}(M). Sometimes we will refer to g(M) as the geometric rank and to r(M) as the algebraic rank of M.

F. Waldhausen [8] asked whether the converse inequality also holds, i.e., whether g(M) = r(M). A positive answer would have implied the Poincaré conjecture. First counterexamples however were found by M. Boileau and H. Zieschang [1]. These examples were Seifert fibered manifolds with g(M) = 3 and r(M) = 2. The work of Y. Moriah and J. Schultens [4] further shows that this class extends to higher genus examples, i.e. Seifert manifolds with g(M) = n + 1 and r(M) = n. In [9] a class of graph manifolds was found for which g(M) = 3 and r(M) = 2. The original Boileau-Zieschang examples can be interpreted as a special case of these graph manifolds.

In this note we will show how the phenomenon observed in [9] generalizes and how it can occur multiple times within a single graph manifold. This yields graph manifolds where the difference between the algebraic and the geometric rank is arbitrarily high.

Key words and phrases. Graph manifolds, Heegaard genus, rank.

The first author was supported in part by NSF Grant #0203680. Both authors were supported by the Max-Planck-Institut für Mathematik, Bonn, Germany.

2. Formulation of the main results

Let M be a closed graph manifold. We will always assume that M comes equipped with its characteristic manifold T = T_{M} and a fixed Seifert fibration on every component of \overline{M − T} . Recall that the Seifert fibrations are unique up to isotopy except for components homeomorphic to Q, the Seifert space with base orbifold the disk with two cone points of order 2. The space Q can also be fibered as the orientable circle bundle over the Möbius band. We will refer to the components of \overline{M − T} as the Seifert pieces of M. Recall the the Seifert pieces of M are up to isotopy precisely the maximal Seifert submanifolds of M. We will only work with totally orientable graph manifolds, i.e. orientable graph manifolds whose Seifert pieces have orientable base orbifold. This makes the Seifert fibrations unique up to isotopy on all Seifert pieces.

Let N be a Seifert piece of M. Denote the fiber of N by f. Let T_{1}, …, T_{n} be the boundary components of N and let γ_{i} ⊂ T_{i} be the curve corresponding to the fiber of the Seifert piece L_{i} where L_{i} is the Seifert piece reached by travelling from N transversely through T_{i}. Note that we possibly have N = L_{i}. The maximality of the Seifert piece N guarantees that for all i the intersection number of f with γ_{i} does not vanish.

We then define N̂ to be the manifold N(γ_{1}, …, γ_{n}) obtained from N by performing a Dehn filling with slope γ_{i} at each boundary component T_{i}. It is clear that the Seifert fibration of N can be extended to a Seifert fibration of N̂ as f has non-trivial intersection number with all γ_{i}.

In the following we will denote the base orbifold of a Seifert piece N by O(N). We will denote an orbifold by its topological type with a list of the orders of cone points, where ∞ stands for a boundary component. We will denote the disc by D, the sphere by S^{2}, the annulus by A, the orientable surface of genus g by F_{g} and the projective plane by P^{2}.

Theorem 1. Let M be a closed graph manifold consisting of two Seifert pieces N_{1} and N_{2} glued along T, where O(N_{1}) = F_{g}(r, ∞), O(N_{2}) = D(p, q) with (p, q) = 1 and min(p, q) ≤ 2g + 1 such that the intersection number of the fibers of N_{1} and N_{2} equals 1.

Then π_{1}(M) is generated by 2g +1 elements. Furthermore M admits a Heegaard splitting of genus 2g + 1 if and only if one of the following holds:

1. N_{2} is the exterior of a s-bridge knot with s ≤ 2g + 1 and the fiber of N_{1} is identified with the meridian of N_{2}, i.e. N̂_{2} = S^{3}.

2. N̂_{1} admits a horizontal Heegaard splitting of genus 2g.

[figure: 1; 5; 9; 4]

Figure 1. A graph manifold with 5-generated fundamental group

We will further see that all manifolds of this type admit a Heegaard splitting of genus 2g+2. Furthermore do most do most of these manifolds not admit a Heegaard splitting of genus 2g + 1 as for any given pair of such manifolds N_{1} and N_{2} there are at most three glueing maps that yield a graph manifold of genus 2g + 1. It is furthermore possible to show that π_{1}(M) cannot be generated by less than 2g + 1 elements, the argument however is complicated.

A careful analysis of the above examples shows that the phenomenon is of local nature, it can therefore be reproduced multiple times within a graph manifold with more complex underlying graph. This yields the following:

Theorem 2. For any n ∈ ℕ there exists a graph manifold M_{n} with 3n-generated fundamental group that has Heegaard genus 4g.

This paper is organized as follows. In Section 3 we review the structure theorem for Heegaard splittings of totally orientable graph manifolds as proven in [6]. Then we study in more detail how Heegaard surfaces can intersect the Seifert pieces that are the building blocks of our examples. In Section 5 and Section 6 we will then give the proofs of Theorem 1 and Theorem 2. We conclude by describing a class orientable Seifert manifolds with 2n-generated fundamental group which we believe to be of Heegaard genus 3n. These manifolds are however not totally orientable.

3. Heegaard splittings of totally orientable graph manifolds

A graph manifold M is totally orientable if every Seifert piece N of M fibers over an orientable base space and if M itself is orientable. In [6] it is shown that the Heegaard splittings of totally orientable graph manifolds have a structure that can be completely described. We use the terms horizontal and vertical to describe the possible structure for the restriction of the Heegaard surface to the Seifert pieces of a graph manifold. Note however, that the restriction of the Heegaard surface is typically not horizontal (i.e., everywhere transverse to the fibers) or vertical (everywhere aligned with the fibers) in the true sense.

Let M be a graph manifold and M = V ∪_{S} W be a Heegaard splitting of M. Let N be a Seifert piece of M. We say that S∩N is h1 if it is a horizontal incompressible surface with an even number of components or if it is obtained from a such a surface by adding some number of tubes. Here adding a tube means adding an annulus that is the boundary of a regular neighborhood of a portion of a fiber and removing the two small disks bounded by the ends of the tube.

Let T be a component of T_{M} that meets N. Let N(T) be a collar of T in N. Then ∂N(T) = T_{0} ∪T_{1} (where T_{0} = T). If there are curves c, c′ ⊂ T such that c∩c′ consists of a single point p, and S ∩ N(T) is isotopic to the boundary of a collar in N(T) of (c × {0}) ∪ (c′ × {1}) ∪ (p × I) and so that S ∩ (N − N(T)) is a (two component) horizontal incompressible surface, then we say that S ∩ N is h2.

If there is a regular or exceptional fiber e in N with regular neighborhood η(e), such that e ⊂ S and S ∩ (N − η(e)) is a (two sheeted) horizontal incompressible surface, and such that Sη(e) is an annulus that wraps once around e, then we say that S ∩ N is h3. If S ∩ N is either h1, h2, or h3, we say that S ∩ N is horizontal.

We say that S ∩ N is vertical if it is a separating (possibly disconnected) vertical incompressible surface or if it is obtained from such a surface by adding tubes. Here adding a tube means adding an annulus that is the boundary of a regular neighborhood of a portion of a horizontal arc that projects to an imbedded arc and removing the two small disks bounded by the ends of the tube.

We say that M = V_{1} ∪_{S} V_{2} is standard if for each component N of M − T , S ∩ N is either horizontal or vertical.

The main theorem in [6] is the following:

Theorem 3. Let M be a totally orientable graph manifold. For any Heegaard splitting M = V_{1} ∪_{S} V_{2}, S can be isotoped so that S ∩ N is standard for every Seifert piece N of M.

The most confounding example of these options appears to be h3. But this option is not as esoteric as it appears. In fact horizontal surfaces of type h3 in a Seifert piece are closely related to horizontal Heegaard surfaces of closed Seifert manifolds as studied in [4].

Consider the following: suppose that M is a closed totally orientable graph manifold and that S is a Heegaard surface that on each Seifert piece, S is either horizontal or vertical. This implies in particular that for any torus T ⊂ T_{M} the set S ∩ T consists of a family of parallel simple closed curves. To any Seifert piece N of M we associate the manifold N_{S}, which is the manifold obtained from N by performing a Dehn filling at every component T of ∂N along a slope represented by the curves S ∩ T. We have the following observation:

Lemma 4. Suppose that S ∩ N is of type h3. Then the Seifert manifold S_{N} has a horizontal Heegaard surface S′ such that S′ ∩ N = S ∩ N. If S ∩ N is planar then S′ is homeomorphic to S^{2}. □

Proof. There is clearly only one way to extend S ∩ N to a horizontal Heegaard surface of S_{N} namely by gluing meridional discs of the glued in solid tori to the boundary components of S∩N. Clearly this surface is a horizontal Heegaard surface for S_{N}. If S∩N is planar then all but one boundary component get capped of which results in a disc whose regular neighborhood is a ball. The assertion follows.

4. Some lemmata

The following lemmata will enable us to compute the Heegaard genus of certain graph manifold in the next section. We start by discussing the possible horizontal surfaces of type in h3 in the relevant Seifert manifolds. Some proofs rely on the theory of 2-dimensional orbifolds and their covering theory as discussed in [7].

Lemma 5. Let M be a graph manifold and N be a Seifert piece with O(N) = D(p, q) and (p, q) = 1. Suppose S ∩ N is a planar surface that is horizontal of type h3. Then the following hold:

1. N_{S} is homeomorphic to S^{3}.

2. S ∩ ∂N contains exactly 2p or 2q components. It should be noted that N_{S} being homeomorphic to S^{3} is equivalent to N being the exterior of an r-bridge knot with meridian μ parallel to ∂N ∩S where r = min(p, q).

Proof. Possibly after exchanging p and q we can assume that S is horizontal in the space N̄ obtained from N after removing a regular neighborhood of the exceptional fiber corresponding to the cone point of order q or by removing a neighborhood of a regular fiber. Clearly N̄ is a Seifert space with O(N̄) = A(p) or O(N̄) = A(p, q). Let T_{1} be the boundary component of N̄ that bounds the drilled out solid torus and T_{2} be the boundary of N. Let S̄ be a component to S ∩ N̄. Clearly S̄ is planar as it is a subsurface of a planar surface.

As we assume that S is of type h3 in N it follows that S̄ ∩ T_{1} consists of a single loop α. Let γ be one component of S̄ ∩ T_{2} and let g be an element of π_{1}(N̄) corresponding to γ. Recall that all other components of S̄ ∩ T_{2} are parallel to γ. Let n be the intersection number of γ with the fiber.

As S̄ is horizontal in N̄ it follows that there exists a finite sheeted orbifold covering p : S̄ → O(N̄), in particular p_{∗}(π_{1}(S̄)) is of finite index in π_{1}(O(N̄)). We distinguish the cases O(N̄) = A(p) and O(N̄) = A(p, q).

Case 1: O(N̄) = A(p). We have π_{1}(A(p)) = ⟨x, y|x^{p}⟩ where the generator y corresponds to the boundary curve corresponding to T_{2}. This implies in particular that p_{∗}(g) is conjugate to y^{n}.

As S̄ is planar this implies that π_{1}(S̄) is generated by homotopy classes that correspond to the components of S̄ ∩ T_{2}, i.e. p_{∗}(π_{1}(S̄)) is generated by conjugates of the element y^{n}. Let N(y^{n}) be the normal closure of y in π_{1}(A(p)). Clearly π_{1}(A(p))/N(y^{n}) ≅ ℤ_{n} ∗ ℤ_{p} is infinite unless n = 1. As p_{∗}(π_{1}(S̄)) ⊂ N(y^{n}) this implies that n = 1 as otherwise p_{∗}(π_{1}(S̄)) is contained in a subgroup of infinite index in π_{1}(A(p)) and is therefore of infinite index itself. Thus we can assume that n = 1 and that p_{∗}(π_{1}(S̄)) ⊂ N(y).

Note first that the orbifold covering space S̃ corresponding to N(y) is a orbifold without cone points and is homeomorphic to the (p + 1)-punctured sphere. Denote the corresponding covering map by p̃.

[figure: p̃; 4]

Figure 2. The 4-sheeted covering of A(4) by a 5-punctured sphere

As p_{∗}(π_{1}(S̄)) ⊂ N(y) it follows that there exists a covering p′ : S̄ → S̃ such that p = p̃ ∘ p′. Claim p′ is a homeomorphism. As for both S̄ and S̃ all but one boundary component map onto a curve corresponding to the element y it follows that p′ is a homeomorphism when restricted to any of these boundary components. In particular p′ extends to a covering p′_{#} : S̄_{#} → S̃_{#} where S̄_{#} and S̃_{#} are the spaces obtained from S̄ and S̃ by gluing discs to these boundary components. As S̄_{#} and S̃_{#} are discs it follows that the obtained map is a homeomorphism. Thus the original p′ was a homeomorphism which proves the claim.

The second assertion is now immediate as S ∩ N̄ is obtained from 2 copies of S̄ and identifying two boundary components. All resulting boundary components lie in T_{2}. The first assertion follows from Lemma 4.

Case 2: O(N̄) = A(p, q). We have π_{1}(A(p, q)) = ⟨x, y, z|y^{p}, z^{q}⟩ where the generator x corresponds to the boundary curve corresponding to T_{2}. We see as in the first case that p_{∗}(O(N̄)) lies in the kernel of the map ϕ : π_{1}(A(p, q)) → π_{1}(A(p, q))/N(x^{n}). As π_{1}(A(p, q))/N(y^{n}) ≅ ℤ_{n} ∗ ℤ_{p} ∗ ℤ_{q} is infinite for all n ∈ ℕ this implies that p_{∗}(O(N̄)) is of infinite index in π_{1}(A(p, q)) which contradicts our assumption.

Lemma 6. Let M be a graph manifold and let N be a Seifert piece with O(N) = F_{g}(p, ∞) or O(N) = F_{g}(p, ∞, ∞). Suppose that S ∩ N is horizontal of type h3 and χ(S ∩ N) > −8g or χ(S ∩ N) > −8g − 4, respectively. Then the following hold

1. S ∩ T has two components for every component T of ∂N.

2. S ∩ N extends to a horizontal Heegaard surface of genus 2g of N_{S}.

Proof. We only deal with the case O(N) = F_{g}(p, ∞) the other case is analogous.

Suppose that S∩N is of type h3 with respect to the exceptional fiber or a regular fiber and let N̄ be the space obtained by drilling out the neighborhood of this fiber. Let S̄ be a component of N̄ ∩ S. Recall that S ∩ N is obtained from two copies of S̄ by identifying them along a boundary component. In particular we have that χ(S ∩ N) = 2χ(S̄).

Now S̄ is a finite sheeted covering of O(N̄) where O(N̄) = F_{g}(∞, ∞) or O(N̄) = F_{g}(p, ∞, ∞) depending on what kind of fiber was drilled out. Suppose that the covering is n-sheeted. Note that in the case O(N̄) = F_{g}(p, ∞, ∞) it must hold that n ≥ p as otherwise the covering space must be a orbifold with singularities. Thus we have

χ(S ∩ N) = 2χ(S̄) = 2nχ(O(N̄)).

As χ(O(N̄)) = −2g or χ(O(N̄)) = −2g − 1 + 1/p it follows immediately from the hypothesis on the Euler characteristic that n = 1. Thus O(N̄) = F_{g}(∞, ∞), i.e. the exceptional fiber was drilled out. Assertion (1) is now immediate and (2) follows from the proof of Lemma 4

It will be important that many Seifert manifolds do not admit a horizontal surface of type h3 of small genus indiscriminately of what graph manifold they belong to.

Lemma 7. Let N be a Seifert manifold with O(N) = F_{g}(p, ∞) such that the exceptional fiber has invariant (α, β) with 1 ≤ β < α. Then the following hold:

1. If α = 2 then there exist two slopes γ on ∂N such that N(γ) admits a horizontal Heegaard splitting of genus 2g.

2. If α ≠ 2 and β ∈ {1, α − 1} then exists one slope γ on ∂N such that N(γ) admits a horizontal Heegaard splitting of genus 2g.

3. In all other cases N(γ) has no Heegaard splitting of genus 2g if γ ≠ f.

Proof. If γ is the fiber then N(γ) is not a Seifert manifold. In particular N(γ) admits no horizontal Heegaard splitting as those are only defined for Seifert manifolds. If the intersection number m of γ with the fiber is greater than 1 then M(γ) is a Seifert space with base orbifold F_{g}(p, m) which has no Heegaard splitting of genus 2g by (i) of Proposition 1.4 of [1]. Suppose now that m = 1. Let e ∈ ℤ be the Euler class of the Seifert space. By (iii) of Proposition 1.4 of [1] it follows that N(γ) admits no Heegaard splitting of genus 2g unless β − eα = ±1. It is clear that there exists two values for e such that the equation holds if β = 1 and α = 2, that there exists one solution if β ∈ {1, α − 1} and none otherwise. The corresponding Heegaard splittings are constructed in Section 1.10 of [1]. This proves the assertion.

Lemma 8. Let N be a Seifert manifold with O(N) = D(p, q) and (p, q) = 1. Then N contains no compact planar horizontal surface.

Proof. Suppose that S is a compact planar horizontal surface in N. Then there exists a finite sheeted orbifold covering p : S → D(p, q). As all components of ∂S are parallel on ∂N it follows that there exists a number n ∈ ℕ such the restriction of p to any component of ∂S is a n-sheeted covering. This implies that we can extend p to a orbifold covering p : S^{2} → S^{2}(p, q, n) by gluing a disc to any component of ∂S and a disc with a cone point of order n to D(p, q). If n = 1 this yields a contradiction as S^{2}(p, q, 1) = S^{2}(p, q) is a bad orbifold which admits no covering by a manifold. If n ≠ 1, then S^{2}(p, q, n) must be a spherical orbifold with universal cover the sphere. Moreover, N_{S} is a Seifert manifold with O(N_{S}) = S(p, q, n). As such it is irreducible. This yields a contradiction, as S ⊂ N extends to a horizontal, hence incompressible, sphere in N_{S}.

Lemma 9. Let M be a graph manifold and let N be a Seifert fibered submanifold with O(N) = F_{g}(p, ∞) or O(N) = F_{g}(p, ∞, ∞). If S ∩ N is vertical, then χ(S ∩ N) − |S ∩ ∂N| ≤ −4g − 2 or χ(S ∩ N) − |S ∩ ∂N| ≤ −4g − 4, respectively.

Proof. If S ∩N is vertical, then S ∩N is obtained from saturated annuli A_{1}, …, A_{n} and tori T_{1}, …, T_{k} by performing ambient 1-surgery along arcs β_{1}, …, β_{m} that project to disjoint imbedded arcs disjoint from the projection of A_{1}, …, A_{n} and T_{1}, …, T_{k} except at their endpoints. Thus the projections a_{1}, …, a_{n} of A_{1}, …, A_{n}, the projections t_{1}, …, t_{k} of T_{1}, …, T_{k} and the projections b_{1}, …, b_{m} of β_{1}, …, β_{m} form a graph Γ in F_{g}(p, ∞) or F_{g}(p, ∞, ∞). By isotoping β_{1}, …, β_{m}, if necessary, we may assume that Γ is trivalent. We denote the set of vertices of Γ by V Γ and the set of edges by EΓ. A simple calculation shows that #V Γ = 2n + k + 2m and that #EΓ = n + k + 3m.

Recall that the result of cutting a handlebody along annuli is still a handlebody or an union of handlebodies. The preimage under the projection map of F_{g}(p, ∞)−Γ or F_{g}(p, ∞, ∞) − Γ is the result of cutting this union of handlebodies along disks. Thus, this preimage must also be a union of handlebodies. In particular, each component of this preimage must be handlebody. A component of this preimage is just the preimage of a component of F_{g}(p, ∞)−Γ or F_{g}(p, ∞, ∞)−Γ. The preimage of such a component can only be a handlebody if the component is a disk with at most one cone point. Thus Γ cuts F_{g}(p, ∞) or F_{g}(p, ∞, ∞) into a collection of disks with at most one cone point which we denote by DΓ.

This induces a cell decomposition of F = F_{g}(p, ∞) and F = F_{g}(p, ∞, ∞), respectively, where DΓ provides the 2-cells, V Γ the 0 cells and EΓ together with the 2n components of ∂F − V Γ the 1-cells. This implies that for H the underlying surface of F = F_{g}(p, ∞) and F = F_{g}(p, ∞, ∞)

χ(H) = −2g + 1 = #V Γ − (#EΓ + 2n) + #DΓ

and χ(H) = −2g = #V Γ−(#EΓ+2n)+#DΓ, respectively. Clearly |S ∩∂N| = 2n and χ(S ∩ N) = −2m.

Here A_{1}, …, A_{n}, T_{1}, …, T_{k} is a separating collection of annuli and tori. Thus a_{1}, …, a_{n}, t_{1}, …, t_{k} is a separating collection of arcs. Each component of the complement of Γ in H is contained in a component of the complement of a_{1}, …, a_{n}, t_{1}, …, t_{k} in H. It follows that the number of complementary disks for Γ is at least two. Thus,

−\frac{|S ∩ ∂N|}{2} + \frac{χ(S ∩ N)}{2} + 2 =

−n − m + 2 = (2n + k + 2m) − (3n + k + 3m) + 2 ≤ −2g + 1

or

−\frac{|S ∩ ∂N|}{2} + \frac{χ(S ∩ N)}{2} + 2 =

−n − m + 2 = (2n + k + 2m) − (3n + k + 3m) + 2 ≤ −2g.

Lemma 10. Let M be a graph manifold and let N be a Seifert piece with O(N) = F_{g}(p, ∞) or O(N) = F_{g}(p, ∞, ∞). If S∩N is horizontal of type h1, then χ(S∩N) ≤ −4g − 2 or χ(S ∩ N) ≤ −4g − 2 − 2p, respectively.

Proof. Suppose that H is a horizontal incompressible surface in N that covers regular points of F_{g}(p, ∞) k times. Note that here k ≥ p ≥ 2. By the Riemann-Hurwitz formula, χ(H) = k(−2g + \frac{1}{p}) ≤ p(−2g + \frac{1}{p}) = −2pg + 1 ≤ −4g + 1 or χ(H) = k(−2g −1+ \frac{1}{p}) ≤ p(−2g −1+ \frac{1}{p}) = −2pg −p+1 ≤ −4g −p+1, respectively. Here S ∩ N is obtained from a separating horizontal incompressible surface by ambient 1-surgery. A connected horizontal incompressible surface is non separating, thus S ∩ N is obtained from two parallel horizontal incompressible surfaces by ambient 1-surgery. Thus χ(S ∩N) ≤ 2(−4g +1) = −8g +2 ≤ −4g −4+2 = −4g −2 or χ(S ∩ N) ≤ 2(−4g − p + 1) = −8g − 2p + 2 ≤ −4g − 4 − 2p + 2 = −4g − 2 − 2p, respectively. It follows that g(S∩N) ≥ 2g+2− \frac{|S∩N|}{2} or g(S∩N) ≥ 2g+2+p− \frac{|S∩N|}{2}, respectively.

Lemma 11. Let M be a graph manifold and let N be a Seifert piece with O(N) = F_{g}(p, ∞) or O(N) = F_{g}(p, ∞, ∞). If S∩N is horizontal of type h2, then χ(S∩N) ≤ −4g − 4 or χ(S ∩ N) ≤ −4g − 4 − 2p, respectively.

Proof. If S ∩ N is horizontal of type h2, then there is a boundary component T of N such that S ∩ (N − η(T)) is a separating horizontal incompressible surface. The computation in Lemma 10 applies to this portion of S. Furthermore, S ∩ η(T) consists of two annuli tubed together. Thus χ(S ∩ N) ≤ −4g − 4 or χ(S ∩ N) ≤ −4g−4−2p, respectively. Whence genus(S∩N) > 2g+3− \frac{|S∩∂N|}{2} or genus(S∩N) > 2g + 3 + p − \frac{|S∩∂N|}{2}, respectively.

Lemma 12. Let M be a graph manifold and let N be a Seifert piece with O(N) = D(p, q) with (p, q) = 1.

1. If S ∩ N is horizontal of type h1, then for some l ≥ 1, k ≥ pq, |S ∩ N| ≤ 2l, χ(S ∩ N) ≤ 2kl(−1 + \frac{1}{p} + \frac{1}{q}) and genus(S ∩ N) ≥ 2l.

2. If S ∩ N is horizontal of type h2, then χ(S ∩ N) ≤ 2k(−1 + \frac{1}{p} + \frac{1}{q}) − 2 and genus(S ∩ N) ≥ 3.

3. If S ∩ N is horizontal of type h3, then χ(S ∩ N) ≤ −2min(p, q) + 2. Furthermore, either S ∩ N is as in Lemma 5, or genus(S ∩ N) ≥ 2.

Proof. Suppose that H is a horizontal incompressible surface in N that covers regular points of D(p, q) k times. Clearly k ≥ pq as p and q are coprime. By the Riemann-Hurwitz formula, χ(H) = k(−1 + \frac{1}{q} + \frac{1}{p}). If S ∩ N is horizontal of type h1, then S ∩ N is obtained from 2l parallel surfaces H via ambient 1-surgery. Thus χ(S ∩ N) ≤ 2lk(−1 + \frac{1}{p} + \frac{1}{q}). By Lemma 8, H is non planar, so genus(H ∩ N) ≥ 1. Whence genus(S ∩ N) ≥ 2l.

If S ∩ N is horizontal of type h2, then by the same reasoning as in Lemma 11, χ(S ∩ N) ≤ 2k(−1 + \frac{1}{p} + \frac{1}{q}) − 2 ≤ −2pq + 2p + 2q − 2 and genus(S ∩ N) ≥ 3.

Suppose that e is the exceptional fiber of index q and let N′ = N −η(e). Suppose that H′ is a horizontal incompressible surface in N′ that covers regular points k times. Here k ≥ p. Then χ(H′) = k(−1 + \frac{1}{p}) ≤ −p + 1. Thus if S ∩ N is horizontal of type h3 with respect to e, then χ(S ∩ N) ≤ 2k(−1 + \frac{1}{p}) ≤ −2q + 2. The last comment follows immediately from Lemma 5.

5. The proof of Theorem 1

In order to give the proof of Theorem 1 we will first show that the fundamental groups can in fact be generated by 2g +1 elements end then that only the manifolds listed admit a Heegaard splitting of genus 2g + 1.

Lemma 13. The manifolds described in Theorem 1 have 2g + 1-generated fundamental groups.

Proof. We first recall the presentations of the fundamental groups of N_{1} and N_{2}:

π_{1}(N_{1}) = ⟨a_{1}, b_{1}, …, a_{g}, b_{g}, s, t, f_{1} | R ⟩ with
R = {[a_{1}, f_{1}], …, [a_{g}, f_{1}], [b_{1}, f_{1}], …, [b_{g}, f_{1}], [s, f_{1}], [t, f_{1}],
s^{r} = f_{1}^{β}, [a_{1}, b_{1}] · . . . · [a_{g}, b_{g}]st = f_{1}^{e}}

and

π_{1}(N_{2}) = ⟨x, y, f_{2} | [x, f_{2}], [y, f_{2}], x^{p} = f_{2}^{β_{1}}, y^{q} = f_{2}^{β_{2}}⟩.

As the manifold M is obtained from the manifold N_{1} and N_{2} by identifying their boundary it follows from van Kampen’s theorem that

π_{1}(M) = π_{1}(N_{1}) ∗_{C} π_{1}(N_{2}) with C ≅ ℤ^{2}.

Note that f_{1} = xyf_{2}^{l} for some l ∈ ℤ as we assume that the intersection number between f_{1} and f_{2} is 1. A simple calculation (see [5]) shows that n = min(p, q) conjugates of f_{1} generate a subgroup of π_{1}(N_{2}) that maps surjectively onto the orbifold group π_{1}(D(p, q)). We do however need something slightly stronger: Claim: We can choose elements h_{2}, …, h_{n} ∈ π_{1}(N_{2}) such that

U = ⟨f_{1}, h_{2}f_{1}h_{2}^{−1}, …, h_{n}f_{1}h_{n}^{−1}⟩

maps surjectively onto the base group and that additionally h_{i} ∈ U for 2 ≤ i ≤ n.

Choose k_{i} such that ⟨f_{1}, k_{2}f_{1}k_{2}^{−1}, …, k_{n}f_{1}k_{n}^{−1}⟩ maps surjectively. For any k_{i} choose h_{i} ∈ U and z_{i} ∈ ℤ such that k_{i} = h_{i}f_{2}^{z_{i}}. Clearly such h_{i} and z_{i} exist as we assume that U maps surjectively one π_{1}(D(p, q)) and as the kernel is generated by f_{2}. As f_{1} and f_{2} commute it follows that k_{i}f_{1}k_{i}^{−1} = h_{i}f_{2}^{z_{i}}f_{1}f_{2}^{−z_{i}}h_{i}^{−1} = h_{i}f_{1}h_{i}^{−1}. This clearly implies that U = ⟨f_{1}, h_{2}f_{1}h_{2}^{−1}, …, h_{n}f_{1}h_{n}^{−1}⟩. The claim follows.

Note that U is a subgroup of finite index in π_{1}(N_{2}) and that we can choose the elements h_{i} such that π_{1}(N_{2}) = U if and only if N_{2} is the exterior of a torus knot with meridian f_{1}. It is however always true that π_{1}(N_{2}) = ⟨U, C⟩ as f_{2} ∈ C.

Note further that the subgroup ⟨s, f_{1}⟩ of π_{1}(N_{1}) is generated by a single element g_{0} which corresponds to the core of the solid torus corresponding to the exceptional fiber of N_{1}. It follows that g_{0}^{k} = f_{1} for some k ∈ ℤ. In order to prove the lemma we describe elements g_{1}, …, g_{2g} ∈ π_{1}(M) such that π_{1}(M) = ⟨g_{0}, …, g_{2g}⟩.

Recall that by assumption n + 1 ≤ 2g. Put h_{i} = 1 for n + 1 ≤ i ≤ 2g. We define • g_{i} = h_{i}a_{i} for 1 ≤ i ≤ g

• g_{i} = h_{i}b_{i−g} for g + 1 ≤ i ≤ 2g

Claim: U ⊂ ⟨g_{0}, …, g_{2g}⟩.

To see this it clearly suffices to show that f_{1} and the elements h_{i}f_{1}h_{i}^{−1} lie in ⟨g_{0}, …, g_{2g}⟩ for 1 ≤ i ≤ 2g. Clearly f_{1} ∈ ⟨g_{0}, …, g_{2g}⟩ as f_{1} = g_{0}^{k}. Furthermore h_{i}f_{1}h_{i}^{−1} ∈ ⟨g_{0}, …, g_{2g}⟩ for 1 ≤ i ≤ g as g_{i}g_{0}^{k}g_{i}^{−1} = h_{i}a_{i}f_{1}a_{i}^{−1}h_{i}^{−1} = h_{i}f_{1}h_{i}^{−1}. The same argument shows that g_{i}g_{0}^{k}g_{i}^{−1} = h_{i}f_{1}h_{i}^{−1} for g + 1 ≤ i ≤ 2g. Thus the claim is proven.

As h_{i} ∈ U for 1 ≤ i ≤ 2g this implies that h_{i} ∈ ⟨g_{0}, …, g_{2g}⟩ for 1 ≤ i ≤ 2g and therefore h_{i}^{−1}g_{i} ∈ ⟨g_{0}, …, g_{2g}⟩ for 1 ≤ i ≤ 2g. As h_{i}^{−1}g_{i} = a_{i} for 1 ≤ i ≤ g and h_{i}^{−1}g_{i} = b_{i−g} for g + 1 ≤ i ≤ 2g it follows that all a_{i} and b_{i} lie in ⟨g_{0}, …, g_{2g}⟩. Furthermore both f_{1} and s are powers of g_{0} and lie in ⟨g_{0}, …, g_{2g}⟩. The last generator t can be written as a product in the remaining generators by the last relation. Thus all generators of π_{1}(N_{1}) lie in ⟨g_{0}, …, g_{2g}⟩ which shows that π_{1}(N_{1}) ⊂ ⟨g_{0}, …, g_{2g}⟩. Thus C ⊂ ⟨g_{0}, …, g_{2g}⟩ and therefore π_{1}(N_{2}) = ⟨U, C⟩ ⊂ ⟨g_{0}, …, g_{2g}⟩. This shows that π_{1}(M) = ⟨g_{0}, …, g_{2g}⟩.

Lemma 14. Let M be a closed graph manifold consisting of two Seifert manifolds N_{1} and N_{2} glued along T, where O(N_{1}) = F_{g}(p, ∞), O(N_{2}) = D(p, q) with (p, q) = 1 such that the intersection number of the fibers of N_{1} and N_{2} equals 1. Let S be a Heegaard surface of M and suppose that S is in standard position.

1. Suppose that S ∩ N_{1} is vertical. Further suppose that S ∩ N_{2} is as in Lemma 5 and that it is horizontal with respect to the exceptional fiber e of index p. If q ≤ 2g + 1, then S ∩ N_{2} can be extended to a Heegaard surface S̃ such that S̃ ∩ N_{1} is vertical and such that genus(S̃) = 2g + 1.

2. If S ∩ N_{1} is as in Lemma 6 and S ∩ N_{2} consists of a single (vertical) annulus, then genus(S) = 2g + 1.

3. In all other cases, genus(S) ≥ 2g + 2.

Note that despite the fact that S ∩ N_{1} is vertical, it may have lower Euler characteristic than S̃. So it could be the case that genus(S) > 2g + 1. (Note also that although distinct irreducible vertical Heegaard splittings of Seifert fibered spaces all have the same genus, this is not the case for Seifert fibered submanifolds of graph manifolds. This necessitates the more careful phrasing in option 2.)

Proof. Case 1: S ∩ N_{1} is vertical.

If S ∩ N_{2} is horizontal of type h1, then by Lemma 9

χ(S) = χ(S ∩ N_{1}) + χ(S ∩ N_{2}) ≤ −4g − 2 + |S ∩ T | + χ(S ∩ N_{2}) =

−4g − 2 + (−χ(S ∩ N_{2}) + 2|S ∩ N_{2}| − 2genus(S ∩ N_{2})) + χ(S ∩ N_{2}) =

−4g − 2 + 2|S ∩ N_{2}| − 2genus(S ∩ N_{2})).

By Lemma 12,

−4g − 2 + 2|S ∩ N_{2}| − 2genus(S ∩ N_{2})) ≤
−4g − 2 + 4l − 2(2l) = −4g − 2.

Thus genus(S) ≥ 2g + 2.

If S ∩ N_{2} is horizontal of type h2, we similarly obtain genus(S) ≥ 2g + 3.

Suppose that S ∩N_{2} is horizontal of type h3 with respect to the exceptional fiber e and S ∩ (N_{2} − η(e)) consists of two planar surfaces. I.e., suppose that S ∩ N_{2} is as in Lemma 5. Then we obtain

χ(S) ≤ −4g − 2 + |S ∩ T | + χ(S ∩ N_{2}) =
−4g − 2 + |S ∩ ∂N_{2}| + (−|S ∩ ∂N_{2}| + 2) = −4g.

Thus genus(S) ≥ 2g + 1.

Note that equality is obtained if and only if for Γ, H as in the proof of Lemma 9, H − Γ consists of exactly two disks. It is easy to construct examples for which this is the case as long as |S ∩ T | ≤ 4g + 2. If |S ∩ T | > 4g + 2, then this is impossible. It follows that this Heegaard splitting of genus 2g + 1 exists if and only if min(p, q) ≤ 2g + 1.

Suppose that S ∩N_{2} is horizontal of type h3 with respect to the exceptional fiber e and S ∩ (N_{2} − η(e)) consists of two non planar surfaces. Then we obtain

χ(S) ≤ −4g + |S ∩ T | + χ(S ∩ N_{2}) =
−4g + |S ∩ ∂N_{2}| + (2 − 2genus(S ∩ N_{2}) − |S ∩ ∂N_{2}|) ≤ −4g − 2.

Thus genus(S) ≥ 2g + 2.

Case 2: S ∩ N_{1} is horizontal.

If S ∩ N_{2} is vertical, then χ(S ∩ N_{2}) ≤ 0 by construction. If S ∩ N_{2} is horizontal, then χ(S ∩ N_{2}) ≤ 0 by Lemma 12. Thus if S ∩ N_{1} is horizontal of type h1, then χ(S) = χ(S ∩ N_{1}) + χ(S ∩ N_{2}) ≤ −4g − 2 by Lemma 10. Whence genus(S) ≥ 2g + 2. Similarly, if S ∩ N_{1} is horizontal of type h2, then χ(S) ≤ −4g − 4. Whence genus(S) ≥ 2g + 3. Similarly, if S ∩ N_{1} is horizontal of type h3 and not as in Lemma 6, then χ(S) ≤ −8g − 4 by Lemma 11. Whence genus(S) ≥ 4g + 3.

If S ∩ N_{1} is horizontal of type then S ∩ N_{1} must for Euler characteristic reasons be as in the conclusion of Lemma 6, in particular it must be horizontal of type h3 with respect to the exceptional fiber e. Furthermore, it must be constructed from two parallel surfaces homeomorphic to F_{g}(∞, ∞).

If the boundary component of each of these two surfaces meets T in the fiber f_{2} of N_{2}, then S ∩ N_{2} can be vertical. By adjoining a single vertical annulus that cuts N_{2} into regular neighborhoods of the two exceptional fibers, we obtain a Heegaard splitting of genus 2g + 1.

As all essential properly embedded annuli in N_{2} are vertical and all properly embedded discs in N_{2} are boundary parallel it follows that all other possibilities for N ∩ N_{2} yield a Heegaard surface of genus more than 2g + 2.

To complete the proof of Theorem 1 we merely need to put the pieces together:

Lemma 15. If option 1 occurs in Lemma 14, then N_{2} is a q-bridge knot complement and the fiber of N_{1} is identified with the meridian of N_{2}. If option 2 occurs in Lemma 14, then N̂_{1} admits a horizontal Heegaard splitting of genus 2g.

Proof. This follows from Lemma 5 and Lemma 6 respectively.

6. The proof of Theorem 2

In this section we construct for any n ∈ ℕ a graph manifold M_{n} such that π_{1}(M_{n}) is 3n-generated but that the Heegaard genus of M_{n} is 4n. We denote the graph underlying M_{n} by Γ_{n}. Γ_{n} is a tree on 2n + 1 vertices z, c_{1}, …, c_{n}, d_{1}, …, d_{n} and 2n edges e_{1}, …, e_{n}, f_{1}, …, f_{n} such that c_{i} and d_{i} are the endpoints of e_{i} and that d_{i} and z are the endpoints of f_{i}.

[figure: z; d_{1}; c_{1}; d_{2}; c_{2}; d_{3}; c_{3}]

Figure 3. The tree Γ_{3}

The closed graph manifold M_{n} is then constructed as follows, where we denote the Seifert piece corresponding to a vertex v by N_{v}.

1. The intersection number between the fibers of the adjacent Seifert spaces is 1 at any torus of the JSJ decomposition.

2. O(N_{z}) is a n-punctured sphere with at most one cone point and N̂_{z} = S^{3}.

3. O(N_{d_{i}}) = T^{2}(∞, ∞, 5n) and N_{d_{i}} admits no horizontal surface of type h3 that has genus 2.

4. O(N_{c_{i}}) is of type D(p, q) with p ∈ {2, 3} and (p, q) = 1 but N_{c_{i}} is not homeomorphic to the exterior of a torus knot in S^{3}.

Remark 16. Note that (2) is equivalent to stating that N_{z} is the exterior of a Seifert fibered n component n-bridge link in S^{3}, in particular π_{1}(N_{z}) is generated by the fibers of the N_{d_{i}}. The existence of the spaces N_{d_{i}} satisfying (3) is an immediate consequence of Lemma 7.

The first part of the proof of Theorem 2 is again a simple calculation:

Lemma 17. π_{1}(M_{n}) can be generated by 3n elements.

[figure: 1; 1; 1; 1; 1; 1; 3; 7; 4; 3; 3; 3; 2; 3; 3; 17]

Figure 4. A graph manifold M with g(M) = 12 and r(M) ≤ 9

Proof. The proof is almost identically to the proof of Lemma 13 and we frequently omit explicit calculations if they are identical. Recall that

π_{1}(N_{d_{i}}) = ⟨a_{i}, b_{i}, s_{i}, t_{i1}, t_{i2}, f_{i} | R_{i} ⟩ with

R_{i} = {[a_{i}, f_{i}], [b_{i}, f_{i}], [s_{i}, f_{i}], [t_{i1}, f_{i}], [t_{i2}, f_{i}], s_{i}^{5n} = f_{i}^{β_{i}}, [a_{i}b_{i}]t_{i1}t_{i2}s_{i} = f_{i}^{e_{i}}} where t_{i1} corresponds the the boundary component between N_{d_{i}} and N_{z} and t_{i2}

corresponds to the boundary component between N_{d_{i}} and N_{c_{i}}.

Recall from the proof of Lemma 13 that there exist elements h_{i1}, h_{i2} ∈ π_{1}(N_{c_{i}}) such that U_{i} = ⟨f_{i}, h_{i1}f_{i}h_{i1}^{−1}, h_{i2}f_{i}h_{i2}^{−1}⟩ is a subgroup of finite index in π_{1}(N_{c_{i}}) that maps surjectively onto the fundamental group of O(N_{c_{i}}) and that h_{i1}, h_{i2} ∈ U_{i}.

We will show that π_{1}(M_{n}) is generated by the generators g_{1}, …, g_{3n} defined as follows:

1. g_{i} is the generator of the cyclic group ⟨f_{i}, s_{i}⟩ for 1 ≤ i ≤ n.

2. g_{n+i} = h_{i1}a_{i} for 1 ≤ i ≤ n.

3. g_{2n+i} = h_{i2}b_{i} for 1 ≤ i ≤ n.

Let H = ⟨g_{1}, …, g_{3n}⟩. We show that H = π_{1}(M_{n}).

Note first that π_{1}(N_{z}) ⊂ H as g_{i} ∈ H implies f_{i} ∈ H for 1 ≤ i ≤ n and π_{1}(N_{z}) is generated by the f_{i}. This implies that t_{i1} ∈ H for 1 ≤ i ≤ n.

The same calculation as in the proof of Lemma 13 further shows that U_{i} ⊂ H for 1 ≤ i ≤ n. It follows that a_{i}, b_{i} ∈ H for 1 ≤ i ≤ n. Thus π_{1}(N_{d_{i}}) ⊂ H as π_{1}(N_{d_{i}}) is generated by a_{i}, b_{i}, s_{i}, f_{i}, t_{i_{1}} and s_{i} and f_{i} are powers of g_{i}.

It now further follows that π_{1}(N_{c_{i}}) ⊂ H as π_{1}(N_{c_{i}}) is generated by U_{i} and C_{i} where C_{i} = π_{1}(N_{c_{i}}) ∩ π_{1}(N_{d_{i}}).

To conclude it clearly suffices to establish the following:

Proposition 18. The Heegaard genus of M_{n} is 4n.

Proof. A Heegaard splitting M_{n} = V_{n} ∪_{S_{n}} W_{n} of genus 4n for M_{n} exists that is structured as follows: S_{n} ∩ N_{z} is a planar surface that is horizontal of type h3. It meets each component of ∂N_{z} twice. S_{n} ∩ N_{d_{i}} is vertical and consists of two annuli that are parallel into the component of ∂N_{d_{i}} corresponding to the edge between z and d_{i}, one annulus that is parallel to the component of ∂N_{d_{i}} corresponding to the edge between d_{i} and c_{i}. Two nonparallel tubes are attached to these annuli.

S_{n} ∩ N_{c_{i}} is also vertical and consists of one annulus parallel into ∂N_{c_{i}} with one tube attached.

To see that M_{n} admits no Heegaard splitting of genus less than 4n, proceed analogously to the proof of Lemma 14. The restrictions in the setting here show that in all possible cases,

χ(S ∩ (N_{d_{i}} ∪ N_{c_{i}})) ≤ −4(1) − 4 + |S ∩ T_{z_{i}}|

where T_{z_{i}} is the torus that meets N_{z} and N_{d_{i}}. Furthermore,

χ(S ∩ N_{z}) ≤ 2 − 2 ∑_{i} |S ∩ T_{z_{i}}|.

Thus χ(S) ≤ 2 − 8n. Whence genus(S) ≥ 4n.

7. Some comments on non totally orientable graph manifolds

During the proofs of Theorem 1 and Theorem 2 me make extensive use of the structure theorem for Heegaard splittings of totally orientable graph manifolds [6]. We believe however that similar statements are true for graph manifolds in general. This suggest a more straight forward generalization of the example provided in [9] which are not totally orientable.

It should be noted that the verification that the manifolds constructed in [9] are not of Heegaard genus 2 relies on the classification of 3-manifolds with nonempty characteristic submanifold that have a genus 2 Heegaard splitting as given by T. Kobayashi [3].

Thus we conjecture that the manifolds M_{n} constructed below are of Heegaard genus 3n, the same argument as above show that they can be generated by 2n elements.

The graph underlying the manifold M_{n} is again Γ_{n}, the Seifert piece corresponding to the vertex v is again denoted by N_{v} and the following hold:

1. The intersection number between the fibers of the adjacent Seifert spaces is 1 at any torus of the JSJ decomposition.

2. O(N_{z}) is a n-punctured sphere with at most one cone point and N̂_{z} = S^{3}.

3. O(N_{d_{i}}) = P^{2}(∞, ∞, 5n) and N_{d_{i}} admits no horizontal surface of type h3 that has genus 2.

4. O(N_{c_{i}}) is of type D(2, q) with odd q but N_{c_{i}} is not homeomorphic to the exterior of a 2-bridge knot in S^{3}.

References

1. M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math. 76, 1984, 455-468.

2. W.H. Jaco and P.B. Shalen, Seifert fibered spaces in 3-manifolds Memoirs of the AMS 220, Providence, RI, 1989.

3. T. Kobayashi Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21, 1984, 437–455.

4. Y. Moriah and J. Schultens, Irreducible Heegaard splittings of Seifert fibered manifolds are either vertical or horizontal, Topology, 37, 1998, 1087–1112.

5. M. Rost and H. Zieschang, Meridional generators and plat presentations of torus links, J. Lond. Math. Soc., 35, 1987, 551–562.

6. J. Schultens, Heegaard splittings of graph manifolds, preprint 2002.

7. P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15, 1983, 401–487.

8. F. Waldhausen Some problems on 3-manifolds Proc. Symposia in Pure Math. 32, 1978, 313-

[figure: 1; 1; 1; 1; 1; 1; 2; 7; 4; 3; 3; 3; 2; 3; 2; 17]

Figure 5. A graph manifold M with g(M) = 9 and r(M) ≤ 6?

9. R. Weidmann, Some 3-manifolds with 2-generated fundamental group, preprint 2001.

Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA

Current address: MPIM-Bonn, Vivatsgasse 7, D-53111 Bonn, Germany

E-mail address: jcs@mathcs.emory.edu

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany

E-mail address: Richard.Weidmann@ruhr-uni-bochum.de