Barriers to Topologically Minimal Surfaces

Begin by removing from $H$ all components that are boundary-parallel disks. If nothing remains, then the result follows. Otherwise, the resulting surface (which we continue to call $H$) is still topologically minimal, as it has the same disk complex. Now, let $\alpha$ denote a loop of $\partial H$ that is innermost among all such loops that are inessential on $\partial M$ . Then $\alpha$ bounds a compression $D$ for $H$ that is disjoint from all other compressions. Hence, every maximal dimensional simplex of $\Gamma(H)$ includes the vertex corresponding to $D$. We conclude $\Gamma(H)$ is contractible to $D$, and thus $H$ was not topologically minimal. ∎

If $\Gamma(H)=\emptyset$ then the result is immediate, as any surface obtained by $\partial$-compressing an incompressible surface must also be incompressible.

If $\Gamma(H)\neq\emptyset$ then, by assumption, there is a non-trivial map $\iota$ from an $(n-1)$-sphere $S$ into the $(n-1)$-skeleton of $\Gamma(H)$. Assuming the theorem is false will allow us to inductively construct a map $\Psi$ of a $n$-ball $B$ into $\Gamma(H)$ such that $\Psi(\partial B)=\iota(S)$. The existence of such a map contradicts the non-trivialty of $\iota$.

Note that $\Gamma(H)\subset\Gamma(H;\partial M)$. If $\pi_{n-1}(\Gamma(H;\partial M))\neq 1$ then the result is immediate. Otherwise, $\iota$ can be extended to a map from an $n$-ball $B$ into $\Gamma(H;\partial M)$.

Let $\Sigma$ denote a triangulation of $B$ so that the map $\iota$ is simplicial. If $\tau$ is a simplex of $\Sigma$ then we let $\tau^{\partial}$ denote the vertices of $\tau$ whose image under $\iota$ represent $\partial$-compressions. If $\tau^{\partial}=\emptyset$ then let $V_{\tau}=\Gamma(H)$. Otherwise, let $V_{\tau}$ be the subspace of $\Gamma(H)$ spanned by the compressions that can be made disjoint from the disks represented by every vertex of $\iota(\tau^{\partial})$. In other words, $V_{\tau}$ is the intersection of the link of $\tau^{\partial}$ in $\Gamma(H;\partial M)$ with $\Gamma(H)$.

By Lemma 3.4, when $\tau^{\partial}\neq\emptyset$ then $\Gamma(H/\tau^{\partial})$ is precisely $V_{\tau}$. (More precisely, there is an embedding of $\Gamma(H/\tau^{\partial})$ into $\Gamma(H)$ whose image is $V_{\tau}$.) By way of contradiction, we suppose the homotopy index of $\Gamma(H/\tau^{\partial})$ is not less than or equal to $n-{\rm dim}(\tau^{\partial})$. Thus, $V_{\tau}\neq\emptyset$ and

Push the triangulation $\Sigma$ into the interior of $B$, so that $\rm{Nbhd}(\partial B)$ is no longer triangulated (Figure 1(b)). Then extend $\Sigma$ to a cell decomposition over all of $B$ by forming the product of each cell of $\Sigma\cap S$ with the interval $I$ (Figure 1(c)). Denote this cell decomposition as $\Sigma^{\prime}$. Note that $\iota$ extends naturally over $\Sigma^{\prime}$, and the conclusion of Claim 3.6 holds for cells of $\Sigma^{\prime}$. Now let $\Sigma^{*}$ denote the dual cell decomposition of $\Sigma^{\prime}$ (Figure 1(d)). This is done in the usual way, so that there is a correspondence between the $d$-cells of $\Sigma^{*}$ and the $(n-d)$-cells of $\Sigma^{\prime}$. Note that, as in the figure, $\Sigma^{*}$ is not a cell decomposition of all of $B$, but rather a slightly smaller $n$-ball, which we call $B^{\prime}$.

For each cell $\tau$ of $\Sigma^{\prime}$, let $\tau^{*}$ denote its dual in $\Sigma^{*}$. Thus, it follows from Claim 3.6 that if $\sigma^{*}$ is a cell of $\Sigma^{*}$ that is on the boundary of $\tau^{*}$, then $V_{\sigma}\subset V_{\tau}$.

We now produce a contradiction by defining a continuous map $\Psi:B^{\prime}\to\Gamma(H)$ such that $\Psi(\partial B^{\prime})=\iota(S)$. The map will be defined inductively on the $d$-skeleton of $\Sigma^{*}$ so that the image of every cell $\tau^{*}$ is contained in $V_{\tau}$.

For each $0$-cell $\tau^{*}\in\Sigma^{*}$, choose a point in $V_{\tau}$ to be $\Psi(\tau^{*})$. If $\tau^{*}$ is in the interior of $B^{\prime}$ then this point may be chosen arbitrarily in $V_{\tau}$. If $\tau^{*}\in\partial B^{\prime}$ then $\tau$ is an $n$-cell of $\Sigma^{\prime}$. This $n$-cell is $\sigma\times I$, for some $(n-1)$-cell $\sigma$ of $\Sigma\cap S$. But since $\iota(S)\subset\Gamma(H)$, it follows that $\tau^{\partial}=\sigma^{\partial}=\emptyset$, and thus $V_{\tau}=\Gamma(H)$. We conclude $\iota(\tau)\subset V_{\tau}$, and thus we can choose $\tau^{*}$, the barycenter of $\iota(\tau)$, to be $\Psi(\tau^{*})$.

We now proceed to define the rest of the map $\Psi$ by induction. Let $\tau^{*}$ be a $d$-dimensional cell of $\Sigma^{*}$ and assume $\Psi$ has been defined on the $(d-1)$-skeleton of $\Sigma^{*}$. In particular, $\Psi$ has been defined on $\partial\tau^{*}$. Suppose $\sigma^{*}$ is a cell on $\partial\tau^{*}$. By Claim 3.6 $V_{\sigma}\subset V_{\tau}$. By assumption $\Psi|\sigma^{*}$ is defined and $\Psi(\sigma^{*})\subset V_{\sigma}$. We conclude $\Psi(\sigma^{*})\subset V_{\tau}$ for all $\sigma^{*}\subset\partial\tau^{*}$, and thus

Note that

Since ${\rm dim}(\tau^{\partial})\leq{\rm dim}(\tau)$, we have

Thus

and finally

It now follows from Equation 1 that $\pi_{(d-1)}(V_{\tau})=1$. Since $d-1$ is the dimension of $\partial\tau^{*}$, we can thus extend $\Psi$ to a map from $\tau^{*}$ into $V_{\tau}$.

Finally, we claim that if $\tau^{*}\subset\partial B^{\prime}$ then this extension of $\Psi$ over $\tau^{*}$ can be done in such a way so that $\Psi(\tau^{*})=\iota(\tau^{*})$. This is because in this case each vertex of $\iota(\tau)$ is a compression, and hence $V_{\tau}=\Gamma(H)$. As $\iota(S)\subset\Gamma(H)=V_{\tau}$, we have $\iota(\tau^{*})\subset V_{\tau}$. Thus we may choose $\Psi(\tau^{*})$ to be $\iota(\tau^{*})$. ∎

We first employ Theorem 3.5 to obtain a sequence of surfaces, $\{H_{i}\}_{0=1}^{m}$ in $M$, such that

1. (1)

   $H_{0}=H$

2. (2)

   $H_{i+1}=H_{i}/\tau_{i}$, for some simplex $\tau_{i}\subset\Gamma(H_{i};\partial M)\setminus\Gamma(H_{i})$.

3. (3)

   For each $i$ the topological index of $H_{i+1}$ in $M$ is $n_{i+1}\leq n_{i}-\rm{dim}(\tau_{i})$, where $n_{0}=n$.

4. (4)

   $H_{m}$ has topological index at most $n_{m}$ with respect to $\partial M$.

It follows from Lemma 2.3 that $\partial H_{i}$ contains at least one component that is essential on $\partial M$ for each $i<m$. The boundary of $H_{m}$ is essential if and only if $H_{m}$ is not a collection of $\partial$-parallel disks. But in this case, the surface $H$ was isotopic into a neighborhood of $\partial M$. Hence, it suffices to show that for all $i$, the distance between the loops of $H_{i}\cap F$ and $H_{i+1}\cap F$ is bounded by $3\chi(H_{i+1})-3\chi(H_{i})$.

Let $\mathcal{V}$ and $\mathcal{W}$ denote the sides of $H_{i}$ in $M$. If the dimension of $\tau_{i}$ is $k-1$, then $H_{i+1}$ is obtained from $H_{i}$ by $k$ simultaneous $\partial$-compressions. Hence, the difference between the Euler characteristics of $H_{i}$ and $H_{i+1}$ is precisely $k$.

If $k=1$ then the loops of $\partial H_{i}$ can be made disjoint from the loops of $\partial H_{i+1}$. It follows that

and thus the result follows. Henceforth, we will assume $k\geq 2$.

Let $\{V_{1},...,V_{p}\}$ denote the $\partial$-compressions represented by vertices of $\tau_{i}$ that lie in $\mathcal{V}$, and $\{W_{1},...,W_{q}\}$ the $\partial$-compressions represented by vertices of $\tau_{i}$ that lie in $\mathcal{W}$. We will assume that $\mathcal{V}$ and $\mathcal{W}$ were labelled so that $p\leq k/2$. The loops of $H_{i+1}\cap F$ are obtained from the loops of $H_{i}\cap F$ by band sums along the arcs of $V_{i}\cap F$ and $W_{i}\cap F$. By pushing the loops of $H_{i}\cap F$ slightly into $\mathcal{V}$, we see that they meet the loops of $H_{i+1}\cap F$ at most $4p$ times. That is,

When $F$ is not a torus we measure the distance between curves $\alpha$ and $\beta$ in $F$ in its curve complex. By [Hem01], this distance is bounded above by $2+2\log_{2}|\alpha\cap\beta|$. Hence, the distance between $H_{i}\cap F$ and $H_{i+1}\cap F$ is at most $2+2\log_{2}2k$. But for any positive integer $k$, $\log_{2}2k\leq k$. Hence, we have shown $d(H_{i}\cap F,H_{i+1}\cap F)\leq 2+2k$. Since $k\geq 2$ it follows that $2+2k\leq 3k$, and thus the result follows.

When $F\cong T^{2}$, the distance between curves $\alpha$ and $\beta$ is measured in the Farey graph. In this case their distance is bounded above by $1+\log_{2}|\alpha\cap\beta|$. As this bound is twice as good as before, the distance between $H_{i}\cap F$ and $H_{i+1}\cap F$ must satisfy the same inequality. ∎

There can be at most $1-\chi(H)$ non-parallel essential arcs on $H$, and at most $1-\chi(Q)$ non-parallel essential arcs on $Q$. Hence, if the number of arcs in $H\cap Q$ incident to $F$ is larger than $(1-\chi(H))(1-\chi(Q))$, then there must be at least two arcs $\alpha$ and $\beta$ of $H\cap Q$, incident to $F$, that are parallel on both $H$ and $Q$. Suppose this is the case, and let $R_{H}$ and $R_{Q}$ denote the rectangles cobounded by $\alpha$ and $\beta$ on $H$ and $Q$, respectively. Note that $\alpha$ and $\beta$ can be chosen so that $A=R_{H}\cup R_{Q}$ is an embedded annulus.

Since $\alpha$ is essential on $Q$ and $Q$ is $\partial$-incompressible, it follows that $\alpha$ is essential in $M$. As $\alpha$ is also contained in $A$, we conclude $A$ is $\partial$-incompressible. If $A$ is also incompressible then the result follows, as $A$ meets $H$ in fewer arcs than $Q$, and $A$ is incident to $F$.

If $A$ is compressible then it must bound a 1-handle $V$ in $M$, since it contains an arc that is essential in $M$. The 1-handle $V$ can be used to guide an isotopy of $Q$ that takes $R_{Q}$ to $R_{H}$. This isotopy may remove other components of $Q\cap V$ as well. The resulting surface $Q^{\prime}$ meets $H$ in at least two fewer arcs that are incident to $F$. ∎

By Lemma 3.2 $H$ and $Q$ can be isotoped so that they meet in a collection of loops and arcs that are essential on both surfaces. Assume first $Q$ is not an annulus, and it meets $H$ in the least possible number of essential arcs. Then by Lemma 4.1, the number of arcs in $H\cap Q$ incident to $F$ is at most $(1-\chi(H))(1-\chi(Q))$. As each such arc has at most two endpoints on $F$,

As in the proof of Corollary 3.7, when $F$ is not a torus we measure the distance between curves $\alpha$ and $\beta$ in $F$ in its curve complex. By [Hem01], this distance is bounded above by $2+2\log_{2}|\alpha\cap\beta|$. So we have,

But for any positive integer $n$, $\log_{2}2n\leq n$. Hence,

When $F\cong T^{2}$, the distance between curves $\alpha$ and $\beta$ is measured in the Farey graph. In this case their distance is bounded above by $1+\log_{2}|\alpha\cap\beta|$. As this bound is twice as good as before, the distance between $H\cap F$ and $Q\cap F$ must satisfy the same inequality.

If $Q$ is an annulus, then since $M$ is not an $I$-bundle we may apply Lemma 3.2 of [Li], which says that $Q\cap F$ is at most distance 2 away from $Q^{\prime}\cap F$, for some incompressible, $\partial$-incompressible annulus $Q^{\prime}$ that meets $H$ in the least possible number of essential arcs. By Lemma 4.1, the number of arcs in $H\cap Q^{\prime}$ incident to $F$ is at most $(1-\chi(H))(1-\chi(Q^{\prime}))$. As above, this implies the distance between $H\cap F$ and $Q^{\prime}\cap F$ is at most

It follows that the distance between $H\cap F$ and $Q\cap F$ is at most $4+2(1-\chi(H))(1-\chi(Q))$. ∎

If $H$ is not isotopic into a neighborhood of $\partial M$ then by Corollary 3.7 we may obtain a surface $H^{\prime}$ from $H$ by a sequence of $\partial$-compressions, which is topologically minimal with respect to $\partial M$, where

By Lemma 4.2,

Putting these together gives:

When $\chi(Q)<0$, then the expression $(\chi(H^{\prime})-\chi(H))(1+2\chi(Q))$ is negative. Hence, we have

On the other hand, when $\chi(Q)=0$ then we have

∎

Suppose $H$ is a topologically minimal surface in $M$ whose genus is at most $g$. By Theorem 2.2 $H$ may be isotoped so that it meets $F$ in a collection of saddles, and so that the components of $H\setminus N(F)$ are topologically minimal in $M\setminus N(F)$. Note that $M\setminus N(F)=X^{\prime}$, where $X^{\prime}\cong X$. We denote the images of $F_{-}$ and $F_{+}$ in $X^{\prime}$ by the same names. Let $H^{\prime}=H\cap X^{\prime}$. By Lemma 2.3, $\partial H^{\prime}$ consists of essential loops on $\partial X^{\prime}$. When projected to $F$, these loops are all on the boundary of a neighborhood of the 4-valent graph $H\cap F$, it follows that the distance between $H^{\prime}\cap F_{-}$ and $H^{\prime}\cap F_{+}$ is at most one.

If $H$ could not have been isotoped to be disjoint from $F$, then the surface $H^{\prime}$ can not be isotopic into a neighborhood of $\partial X^{\prime}$. We may thus apply Lemma 4.3 to obtain:

Note that the theorem only has content when $g\geq 2$, since by definition a torus cannot be topologically minimal. Also, $Q$ has non-empty boundary, so $1-\chi(Q)\geq 1$. It follows that

∎