Heegaard splittings of 3–manifolds (Haifa 2005)
Problems

We analyze what is known, and what is not known, about the following question:

The only known examples of 3–manifolds that admit infinitely many Heegaard splittings of the same genus are given in Sakuma [35], Morimoto and Sakuma [28] and Bachman and Derby-Talbot [3]. By Li’s proof of the Waldhausen conjecture [19], any such manifold must have an essential torus $T$. In the presence of such a torus, it is easy to see how an infinite number of Heegaard splittings of the same genus might arise. Simply take any splitting that intersects $T$, and Dehn twist the splitting about $T$. But is this the only way such an infinite collection might arise? What can be said about the manifold if this construction does not produce an infinite collection of non-isotopic splittings?

We analyze what is known when the torus is separating. (Some of the results described below hold in the non-separating case as well.) Suppose then $T$ is an essential torus which separates a closed, orientable, irreducible 3–manifold $M$ into $X$ and $Y$. Then we may think of $M$ as being constructed from $X$ and $Y$ by gluing by some homeomorphism, $\phi\co\partial X\to\partial Y$.

Fix separate triangulations of $X$ and $Y$ (these do not have to agree in any way on $T$). By a result of Jaco and Sedgwick [13] the sets $\Delta_{X}$ and $\Delta_{Y}$ of slopes on $\partial X$ and $\partial Y$ that bound normal or almost normal surfaces in $X$ and $Y$ are finite (see Bachman [2] for a further discussion of the almost normal case). We may thus define the distance $d(T)$ of $T$ to be the distance between the sets $\phi(\Delta_{X})$ and $\Delta_{Y}$, as measured by the path metric in the Farey graph of $\partial Y$.

The relevance of this theorem is that Dehn twisting an amalgamation about the torus $T$ will not produce non-isotopic Heegaard splittings. Hence, when $d(T)\geq 2$ the manifold $M$ can only admit an infinite collection of non-isotopic Heegaard splittings if $X$ or $Y$ does. The converse, however, is more subtle. That is, if we know $X$ or $Y$ has infinitely many non-isotopic Heegaard splittings of the same genus, then does it follow that $M$ does as well? We conjecture the following:

When $d(T)=0$ or $1$, there is the possibility that there is a strongly irreducible Heegaard splitting of $M$. Let $H$ denote such a splitting surface. By a classic result of Kobayashi, $H$ can be isotoped to meet $T$ in a non-empty collection of loops that are essential on both surfaces.

One way Dehn twisting $H$ about $T$ can fail to produce a non-isotopic Heegaard splitting is if $H\cap X$ is a fiber of a fibration of $X$. Then the effect of the Dehn twist can be “undone” by pushing $H\cap X$ around the fibration. This is precisely what happens when $T$ is a separating vertical torus in a Seifert Fibered space. (For a complete resolution of \fullrefMainQuestion in the case of Seifert Fibered spaces, see [3].) A second thing that may happen is that $H\cap X$ is the union of two pages of an open book decomposition of $X$. Then the effect of the Dehn twist can be undone by spinning $H\cup X$ about the open book decomposition. We conjecture that these are the only ways that Dehn twisting $H$ about $T$ can fail to produce non-isotopic Heegaard splittings:

Let $F$ be a closed surface of genus $1$ standardly embedded in $S^{3}$, that is, it bounds a solid torus on each of its sides. We say that a knot $K$ has a $(1,b)$–presentation or that it is in a $(1,b)$–position, if $K$ has been isotoped to intersect $F$ transversely in $2b$ points that divide $K$ into $2b$ arcs, so that the $b$ arcs in each side can be isotoped, keeping the endpoints fixed, to disjoint arcs on $F$. The genus–1–bridge number of $K$, $b_{1}(K)$, is the smallest integer $n$ for which $K$ has a $(1,n)$–presentation. We say that a knot is a $(1,n)$–knot if $b_{1}(K)\leq n$. If $K$ is a $(1,1)$–knot, then it is easy to see that $K$ has tunnel number one. On the other hand, if $K$ has tunnel number one, it seems to be very difficult to determine $b_{1}(K)$. It has been of interest to find tunnel number one knots with large genus–1–bridge number, see for example Kobayashi and Rieck [16].

Moriah and Rubinstein [24] showed the existence of tunnel number one knots $K$ with $b_{1}(K)\geq 2$. Morimoto, Sakuma and Yokota also showed this, and gave explicit examples of knots $K$, with tunnel number one and $b_{1}(K)=2$ [29]. It was shown by Eudave-Muñoz [7], that many of the tunnel number one knots $K$ constructed in [5] are not $(1,1)$–knots; this was extended in [8], where it is shown that many such knots are not $(1,2)$–knots. We remark that such knots can be explicitly described, for example the knot $K$ shown in Figure 13 of [8] satisfies $3\leq b_{1}(K)\leq 4$. Combining results of [6] and [9], it is also possible to give explicit examples of knots $K$, with tunnel number one and $b_{1}(K)=2$, for example the knot shown in Figure 4 of [6]. Valdez-Sánchez and Ramírez-Losada have also shown explicit examples of tunnel number one knots $K$ with $b_{1}(K)=2$ (personal communication). These knots bound punctured Klein bottles but are not contained in the $(1,1)$–knots bounding Klein bottles determined by the same authors [33].

Johnson and Thompson [14], and independently Minsky, Moriah and Schleimer [22] have shown that for any given $n$, there exist tunnel number one knots which are not $(1,n)$–knots. The two papers use similar techniques, prove the existence of such knots, but do not give explicit examples. For a tunnel number one knot, let $\Sigma$ be the Heegaard splitting of the knot exterior determined by the unknotting tunnel, and let $d(\Sigma)$ denote the Hempel distance of the splitting [12]. In [14] and [22] the existence of tunnel number one knots with large Hempel distance is shown, and then results of Scharlemann and Tomova [37], [40] are used to ensure the knots have large genus–1–bridge number.

We propose the following problems:

Let $M_{1}$ and $M_{2}$ be two manifolds with connected boundary $\partial M_{1}\cong\partial M_{2}\cong S$ and $\phi\co\partial M_{1}\to\partial M_{2}$ a homeomorphism. If one glues $M_{1}$ to $M_{2}$ by identifying $x$ to $\phi(x)$ for each $x\in\partial M_{1}$, one obtains a closed manifold $M$. We say that $M$ is an amalgamation of $M_{1}$ and $M_{2}$.

Let $\mathcal{D}_{i}$ ($i=1,2$) be the set of curves in $\partial M_{i}$ that bound essential disks in $M_{i}$. We propose the following conjecture.

Camal is a generalization of two recent theorems. In the case that $M_{1}$ and $M_{2}$ are atoroidal and have incompressible boundaries, ie, $\mathcal{D}_{1}=\mathcal{D}_{2}=\emptyset$, the conjecture is proved by Souto [39] and Li [20]. Note that [20] also gives an algorithm to find $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$.

In the case that both $M_{1}$ and $M_{2}$ are handlebodies, $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ can be chosen to be empty and \fullrefCamal follows from a recent theorem of Scharlemann and Tomova [37]. The theorem of Scharlemann and Tomova can be formulated as: if the distance between $\mathcal{D}_{2}$ and $\phi(\mathcal{D}_{1})$ (ie, the Hempel distance) is large, then the genus of any other Heegaard splitting must be large unless it is a stabilized copy of $S$.

For a handlebody $H$ we have an inclusion of mapping class groups, $MCG(H)<MCG(\partial H)$. If $M=H_{+}\cup_{S}H_{-}$ is a Heegaard splitting we denote $\Gamma_{\pm}=MCG(H_{\pm})$ $<MCG(S)$, and moreover let $\Gamma^{0}_{\pm}$ be the kernel of the map $MCG(H_{\pm})\to$${\rm Out}(\pi_{1}(H_{\pm}))$ (ie, $\Gamma^{0}_{\pm}$ is the group of mapping classes of $H_{\pm}$ that are homotopic to the identity on $H_{\pm}$).

When $M=S^{3}$ and ${\rm genus}(S)=2$, Akbas [1] proved $\Gamma_{+}\cap\Gamma_{-}$ is finitely presented. (Finitely generated has a longer history starting with Goeritz [11] – for details see Scharlemann [36].)

Let $\Delta_{\pm}\subset{\cal C}(S)$ be the set of (isotopy classes of) simple curves in $S$ which bound disks in $H_{\pm}$. Let $Z\subset{\cal C}(S)$ be the simple curves in $S$ that are homotopic to the identity in $M$. Note that $Z$ contains $\Delta_{\pm}$, and is invariant under $\Gamma^{0}_{\pm}$.

Namazi [30] showed if the distance of the splitting (${\rm dist}_{{\cal C}(S)}(\Delta_{+},\Delta_{-})$) is sufficiently large then $\Gamma_{+}\cap\Gamma_{-}$ is finite. Note also the Geometrization Theorem plus Hempel [12], plus Mostow rigidity plus Gabai–Meyerhoff–Thurston [10], implies that the image of $\Gamma_{+}\cap\Gamma_{-}$ in $MCG(M)$ is finite when the splitting distance is at least 3.

Remarks\quaWhen $M=S^{3}$ or a connected sum of $S^{2}\times S^{1}$’s, with the natural splitting, equality holds trivially. If $M$ is a lens space $L_{p,q}$, with $p\geq 2$, and $S$ is the torus, then they are not equal – $Z$ is all of ${\cal C}(S)=\Q\cup\{\infty\}$, whereas the orbit of $\Delta_{+}\cup\Delta_{-}=\{\frac{0}{1},\frac{p}{q}\}$ under $\langle\Gamma^{0}_{+},\Gamma^{0}_{-}\rangle<SL(2,\Z)$ is strictly smaller than $\Q\cup\{\infty\}$. One might reasonably ask if there is equality when $M$ is hyperbolic, or if the splitting distance is sufficiently large.

There are plenty of examples for manifolds with strongly irreducible Heegaard splittings of arbitrarily high genus. However there are no examples of manifolds (closed or with a single boundary component) with a weakly reducible but non-stabilized non-minimal Heegaard splitting.

Recently T Kobayashi and Y Rieck disproved the conjecture for knots which are not prime (see [17]). The conjecture is known by work of Morimoto for tunnel number one knots and for knots which are connected sums of two prime knots each of which is also $m$–small. (See [23] for more references.)

In [21] Lustig and Moriah defined a condition on complete disks systems in a Heegaard splitting, called the double rectangle condition. It was shown that if a manifold has a Heegaard splitting which has some complete disk system which satisfies this condition then there are only finitely many such disk systems with that property. However the double rectangle condition is clearly non-“generic”. Is it possible to define some other condition which will be “generic” in some reasonable sense? The intuition is that if the Heegaard distance of the splitting is sufficiently high then some form of this might be possible.

Let $(M;V,W)$ be a Heegaard splitting of genus $n$. Let $g_{1},\ldots,g_{n}\in\pi_{1}(M)$ be the elements corresponding to a spine of $V$.

(This would be a kind of “Freiheitsatz”.) For instance, what happens with the Casson–Gordon examples of strongly irreducible splittings of a fixed $M$ of arbitrarily high genus?

Note that the freeness holds for the examples exhibited by Namazi [31] and Namazi–Souto [32].

Here a $n$–tuple is called irreducible if it is not Nielsen equivalent to a tuple of type $(x_{1},\ldots,x_{n-1},1)$. Note that free groups and free Abelian groups have this property, in fact for those groups there is precisely one irreducible Nielsen equivalence class. Note that for Heegaard splittings it has been shown by Tao Li that closed non-Haken 3–manifolds only have finitely many isotopy classes of irreducible Heegaard splittings so those groups might be potential examples.