2 Intersection norms and Thurston’s construction of 3-manifolds realizing polygons

In this section, we first recall some basic facts about intersection norms on closed oriented surfaces; see [Reference Cossarini and Dehornoy2] for more details. We finish by explaining the idea in Thurston’s proof of Theorem 1 from which one can see our generalization.

2.1 Intersection norms:

They are integer norms defined on the first homology of a closed oriented surface $\Sigma _g$ . Introduced by Turaev in [Reference Turaev10] intersection norms received a new interpretation in the article of Cossarini and Dehornoy [Reference Cossarini and Dehornoy2]: they used intersection norms to classified Birkhoff sections of the geodesic flow on the unit tangent bundle of a closed oriented surface.

Let $\Gamma =\{\gamma _1,...,\gamma _n\}$ be a finite collection of closed curves on $\Sigma _g$ with only transverse intersection points. Assume that $\Gamma $ is a filling collection , i.e., its complement in $\Sigma _g$ is a union of topological disks. The function

$$ \begin{align*} N_{\Gamma}:H_1(\Sigma_g,\mathbb{Z})&\longrightarrow \mathbb{N}\\[-2pt] a&\longmapsto \inf\{\mathrm{card}\{\alpha\cap\Gamma\}; [\alpha]=a \}, \end{align*} $$

where $\alpha $ is an oriented collection of closed curves representing a with each of its components transverse to $\Gamma $ , satisfies the following properties:

1. – seperation: $N_{\Gamma }=0$ if and only if $a=0$ —since $\Gamma $ is filling;

2. – linearity on rays: $N_{\Gamma }(n.a)=n.N_{\Gamma }(a)$ for $a\in H_1(\Sigma _g,\mathbb {Z})$ and $n\in \mathbb {N}$ ; and

3. – convexity: $N_{\Gamma }(a+b)\leq N_{\Gamma }(a)+N_{\Gamma }(b)$ for $a;b\in H_1(\Sigma _g,\mathbb {Z})$ .

In the definition of $N_{\Gamma }$ , $\Gamma $ is fixed in its homotopy class and $N_{\Gamma }$ computes the minimal intersection number with $\Gamma $ among all the representatives of a homology class.

For $n\in \mathbb {N}^*$ and $a\in H_1(\Sigma _g,\mathbb {Z})$ , we set $N_{\Gamma }(\frac {1}{n}.a):=\frac {1}{n}N_{\Gamma }(a)$ and by linearity on rays, $N_{\Gamma }$ extends to a well-defined function on $H_1(\Sigma _g,\mathbb {Q})$ . In fact, $N_{\Gamma }(\frac {n}{n}.a)=\frac {1}{n}N_{\Gamma }(n.a)=\frac {1}{n}(n.N_{\Gamma }(a))=N_{\Gamma }(a)$ . By density, $N_{\Gamma }$ extends to a norm on $H_1(\Sigma _g,\mathbb {R})$ called the intersection norm .

By definition, $N_{\Gamma }$ is also an integer norm. Therefore, its dual unit ball is the convex hull of finitely many vectors $v_i\in H^1(\Sigma _g,\mathbb {Z})$ . Like dual unit balls of Thurston norms on 3-manifolds with toral boundary components, the vectors $v_i$ also satisfy the parity condition.

We recall that the norm is completely determined by the vectors $v_i$ :

$$ \begin{align*}N_{\Gamma}(a)=\displaystyle{\max_{v_i}\{\langle v_i,a\rangle\}}.\end{align*} $$

Cossarini and Dehornoy provided an algorithm that computes all the vectors of the dual unit ball of an intersection norm. It is also known that symmetric integer polygons satisfying the parity condition are dual unit balls of intersection norms (see [Reference Sane7]-Proposition 9), but there are examples of such polytopes in dimension $4$ that cannot be realized by intersection norms. Here is the class of polytopes that interest us.

Definition 1 A filling collection $\Gamma $ on $\Sigma _g$ is homologically nontrivial if there exists an orientation $\overset {\rightarrow }{\Gamma }$ of $\Gamma $ such that $[\overset {\rightarrow }{\Gamma }]$ is a nontrivial homology class. A homologically nontrivial polytope in $\mathbb {Z}^{2g}$ is a symmetric polytope, satisfying the parity condition, that appears like the dual unit ball of an intersection norm on $\Sigma _g$ associated to a homologically nontrivial collection.

A filling collection $\Gamma $ is homologically nontrivial if and only if at least one of its component is homologically nontrivial. So, many filling collections on $\Sigma _g$ are homologically nontrivial and therefore, most of dual unit balls of intersection norms are homologically nontrivial.

2.2 Thurston’s construction:

Let $\Gamma =\{\gamma _1,...,\gamma _n\}$ be a filling collection of closed geodesics on the flat torus $\mathbb {T}$ . Since every component of $\Gamma $ is simple and nonseparating, there is an orientation of each component of $\Gamma $ making the oriented collection $\overrightarrow {\Gamma }$ nontrivial in homology: every collection of geodesics on the torus is homologically nontrivial.

By applying the operation on Figure 1 at finitely many well-chosen double points, we obtain a filling closed curve $\overrightarrow {\gamma }$ on $\mathbb {T}$ which is no longer a geodesic.

Figure 1 Attaching two curves at an intersection point.

Now, let $\pi :M\longrightarrow \mathbb {T}$ be the circle bundle over $\mathbb {T}$ with Euler number $1$ . Then, $H_2(M;\mathbb {Z})$ is isomorphic to $H_1(\mathbb {T};\mathbb {Z})$ .

Let K be a lift of $\overrightarrow {\gamma }$ in M and $M_K$ the complement in M of a tubular neighborhood $T(K)$ of K. The morphism

$$ \begin{align*} r:H_2(M;\mathbb{Z})&\longrightarrow H_2(M_K,\partial M_K;\mathbb{Z})\\ [S]&\longmapsto [S\cap M_K] \end{align*} $$

is an isomorphism. In fact, we have the following exact sequence:

$$\begin{align*}0\rightarrow H_2(M;\mathbb{Z})\rightarrow H_2(M,T(K);\mathbb{Z})\rightarrow H_1(T(K);\mathbb{Z})\rightarrow H_1(M;\mathbb{Z}).\end{align*}$$

Since $[\overrightarrow {\gamma }]=\pi _*(K)$ is nonzero, the inclusion $H_1(T(K);\mathbb {Z})\rightarrow H_1(M;\mathbb {Z})$ is injective. It follows that the map $H_2(M;\mathbb {Z})\rightarrow H_2(M,T(K);\mathbb {Z})$ is an isomorphism. By excision, we obtain the isomorphism r.

Thus, $H_2(M_K,\partial M_K;\mathbb {Z})$ is isomorphic to $H_1(\mathbb {T};\mathbb {Z})$ and canonical representatives of $H_2(M_K,\partial M_K;\mathbb {Z})$ are of the form $\pi ^{-1}(\alpha )\cap M_K$ , where $\alpha $ is an oriented simple curve in $\mathbb {T}$ . Since $\pi ^{-1}(\alpha )$ is a torus, the Euler characteristic of $\pi ^{-1}(\alpha )\cap M_K$ is given by its number of boundary components:

$$ \begin{align*}-\chi(\pi^{-1}(\alpha)\cap M_K)=\mathrm{card}\{\pi^{-1}(\alpha)\cap K\}=\mathrm{card}\{\alpha\cap\Gamma\}.\end{align*} $$

Thurston showed that if $\alpha $ minimally intersects $\Gamma $ , then $\pi ^{-1}(\alpha )\cap M_K$ is norm-minimizing:

(1) $$ \begin{align} x([\pi^{-1}(\alpha)\cap M_K])=\displaystyle{\sum_{m=1}^n{i(\alpha,\gamma_m)}}. \end{align} $$

The technical part in Thurston’s proof is the construction of a foliation on $M_K$ without Reeb component and having $\pi ^{-1}(\alpha )\cap M_K$ as a leaf—which by Thurston’s characterization of norm-minimizing surface implies that $\pi ^{-1}(\alpha )\cap M_K$ realizes the norm in its homology class.

Equation (1) describes exactly an equality between Thurston norm on $M_K$ and the intersection norm on the torus associated to $\Gamma $ and this remark is from us. It can be rewritten as follows:

(2) $$ \begin{align} x(a)=N_{\Gamma}(\pi_*(a)). \end{align} $$

Polygons satisfying the parity conditions can be realized as dual unit balls of intersection norms on the torus (see [Reference Sane7, Proposition 9] for the proof of this fact). Equation 2 implies that they can also be realized as dual unit ball of Thurston norms.

We aim to extend Thurston’s construction to higher genus surfaces using intersection norms, namely for every circle bundle $\pi :M\longrightarrow \Sigma _g$ with Euler number equal to 1. For the general case, there are essentially two differences.

1. – There exists filling collections that are not homologically nontrivial. A consequence of this fact is that the dimension of $H_2(M_K)$ increases by one with one homology class corresponding to K. For instance, filling collections made with separating simple closed curves are homologically trivial.

2. – There are examples of filling collections $\Gamma $ and $N_{\Gamma }$ -minimizing oriented curves $\alpha $ for which $\pi ^{-1}(\alpha )\cap N_K$ is not minimizing for the Thurston norm which contrasts with the case of the torus (see Figure 2).

   

   Figure 2 The vertical surface $S:=\pi ^{-1}(\alpha )\subset M_K$ (in the middle) is a torus with four boundary components. By replacing these four boundary components by a handle, we obtain a genus $2$ closed surface $S'$ (on the right-picture) and $|\chi (S')|<|\chi (S)|$ .

3 Incompressible surfaces in nontrivial knot complements in circle bundles

This section is devoted to the study of incompressible surfaces in the complement of a knot in a circle bundle over a closed surface.

If S is compressible in M, then one can cut S along a compression disk (see Figure 3). This cutting operation reduces the complexity of the surface. Therefore, a norm-minimizing surface with nonzero Euler characteristic is incompressible.

Figure 3 Cutting a surface S along a compression disk. This cutting operation reduces the genus by one.

Classification (up to isotopy) of incompressible surfaces of 3-manifolds is an interesting question in topology. For the case of circle bundles over closed surfaces, a complete answer is given in [Reference Waldhausen11].

A circle bundle M over a closed surface is obtained as follows. Let $\Sigma _{g,1}$ be a closed surface with one boundary component and $M':=\Sigma _{g,1}\times \mathbb {S}^1$ be the trivial circle bundle. The bundle structure on $M'$ induces a foliation by vertical closed curves F on its boundary component which is a torus, and let $\alpha $ be the trace of a section $\Sigma _{g,1}\times \{*\}$ of $M'$ on its boundary. Let $\mathbb {D}^2\times \mathbb {S}^1$ be a solid torus, $l:=\{*\}\times \mathbb {S}^1\subset \partial (\mathbb {D}^2\times \mathbb {S}^1)$ be its longitude and $m:=\partial \mathbb {D}^2\times \{*\}$ be its meridian. We obtain a closed 3-manifold M by Dehn-filling $M'$ with $\mathbb {D}^2\times \mathbb {S}^1$ and the bundle structure of $M'$ extend to M if and only if the meridian m is mapped to a curve $\beta \in \partial M'$ which intersects F exactly one time. The geometric intersection between $\alpha $ and $\beta $ is called the Euler number of the circle bundle M. All circle bundles over a closed surface are obtained in this way and the Euler number classified them (see [Reference Montesinos6]).

When the Euler number of M is equal to $1$ , then $\beta =F+\alpha $ . It implies that $[F+\alpha ]=0$ in M. Since $\alpha =\partial (\Sigma _{g,1}\times \{*\})$ , we obtained that $[F]=0$ . So, $\pi _*:H_1(M;\mathbb {Z})\longrightarrow H_1(\Sigma _g;\mathbb {Z})$ is an isomorphim.

One can check the proof of Waldhausen’s theorem in Hatcher’s notes [Reference Gabai4, Propositions 1–11].

The existence of horizontal surfaces in M depends on its Euler number. More precisely, a circle bundle admits a horizontal surface if and only if its Euler number is zero [Reference Gabai4, Proposition 2.2].

We push Waldhausen’s classification a bit further. Let $\pi :M\longrightarrow \Sigma _g$ be a circle bundle with Euler number 1 and K an oriented knot in M such that $\pi (K)$ is a nontrivial homology class. We denote by $M_K$ the complement in M of a tubular neighborhood of K. Let S be a surface embedded in $M_K$ .

The closure $\bar {S}$ is embedded in M and $S=\bar {S}\cap M_K$ . So, to classify incompressible surfaces in $M_K$ , all we need is to understand their closure in M.

For the proof of Main Theorem, we show the following elaborate version of Theorem 2 in the introduction:

Theorem 4 shows that the only obstruction for an incompressible surface to be vertical comes from attaching handles like in Figure 2.

Definition 3.1 Let S be an incompressible surface in $M_K$ and $\alpha $ an essential simple curve on $\bar {S}$ which bounds a disk $\mathbb {D}_{\alpha }$ in M. The weight of $\alpha $ is the integer $w(\alpha )$ defined by:

$$ \begin{align*}w(\alpha)=\min\{\mathrm{card}\{\mathbb{D}_{\alpha'}\cap K\}, \alpha' \hspace{0,1cm} \rm{isotopic\hspace{0,1cm} to}\hspace{0,1cm} \alpha\}\end{align*} $$

The verticality defect of S is the integer $vd(S)$ defined by:

$$ \begin{align*}vd(S)=\max_{\alpha}\{w(\alpha)\}.\end{align*} $$

One can see that $vd(S)$ is equal to zero if and only if S is a vertical surface up to isotopy namely

$$ \begin{align*}S=\pi^{-1}(\alpha)\cap M_K,\end{align*} $$

where $\alpha $ is a simple closed curve on $\Sigma _g$ . Moreover, if $vd(S)=1$ , then S is homologous to a vertical surface with the same Euler characteristic. In fact, if $\alpha $ is a simple curve on $\bar {S}$ such that $w(\alpha )=1$ , we can cut $\bar {S}$ along $\alpha $ to obtain a surface $\bar {S}_1$ . The surface $S_1:=\bar {S}_1\cap M_K$ has two more boundary components than S and one handle less and is homologous to S. It follows that $\chi (S)=\chi (S_1)$ . Repeating this process, we obtain a vertical surface $S_n$ in the same homology class and with the same Euler characteristic like S.

We end this section with some definitions. Let A and B be two sub-arcs of K such that $a:=\pi (A)$ and $b:=\pi (B)$ are disjoint simple arcs with extremities $\partial a=\{t,x\}$ and $\partial b=\{y,z\}$ . Let $\lambda _1$ and $\lambda _2$ be two arcs from t to y and x to z, respectively, such that $\lambda _1$ , a, $\lambda _2$ and b bound a topological disk.

The oriented arcs a and b can be seen as sections of the unit tangent bundle of their supports, and there is a homotopy (see Figure 4) of sections $s_t$ such that:

1. – $s_t$ is an isotopy between the supports of a and b, with extremities gliding in $\lambda _1$ and $\lambda _2$ and

2. – $s_0=a$ and $s_1=b$ .

Figure 4 Rectangle between two arcs obtained by lifting a homotopy between two sections. On the left, we have the case where the orientations of the arcs agree and on the right we have the case where the orientations are opposite.

The isotopy $s_t$ lifts to a rectangle R from A to B and when we blow up R—the blow up of a rectangle R consists in replacing R by the boundary of a tubular neighborhood of R (see Figure 5)— , we obtain a handle (homeomorphic to $\mathbb {S}^1\times [0,1]$ ) enclosing A and B.

Figure 5 Blowing up of a rectangle.

The construction described above works for more than two subarcs and in what follows, we will consider handles as blow up of rectangles between subarcs.

4 Proof of Main theorem

Let us start this section with the following statement: if two filling collections $\Gamma $ and $\Gamma '$ differ by an “attachment” (see Figure 1), then the intersection norm associated to $\Gamma $ is equal to the one associated to $\Gamma '$ (see [Reference Sane7, Lemma 11]). Therefore, any intersection norm is realized by a filling curve $\gamma $ , not necessarily in minimal position.

Let $\overrightarrow {\gamma }$ be an oriented filling curve with non vanishing class in homology. Let $\pi :M\longrightarrow \Sigma _g$ be a circle bundle with Euler number equal to $1$ . Then, $H_2(M;\mathbb {Z})$ is isomorphic to $H_1(\Sigma _g;\mathbb {Z})$ . In fact, by Poincaré duality, $H_2(M;\mathbb {Z})$ is isomorphic to $H_1(M;\mathbb {Z})$ which is isomorphic to $H_1(\Sigma _g,\mathbb {Z})$ since the Euler number is equal to $1$ . Instead of only taking a lift of $\overrightarrow {\gamma }$ in M, we add the modification depicted in Figure 6 on the neighborhood of each fiber of a double point of $\gamma $ . Let $\hat {K}$ be the obtained knot and $M_{\hat {K}}$ be the complement of $\hat {K}$ . Since $\pi (\hat {K})$ is still homologous to $\overrightarrow {\gamma }$ , one can use exact sequence and excision theorems in homology, like in Thurston’s proof (Section 2), to check that $H_2(M_{\hat {K}};\mathbb {Z})$ is isomorphic to $H_1(\Sigma _g;\mathbb {Z})$ with vertical surfaces as canonical representatives.

Figure 6 Modification of K around a fiber of a double point of $\gamma $ . The red arc is coming out of the page. Each vertical arc (the dark and the red one) individually follows a fiber and is linked to itself. Along the fiber, the modified arcs form a braid with two strands twisted three times.

Thurston’s construction does not extend in a trivial way to higher genus surfaces since a norm-minimizing surface S could have verticality defect greater than two (see Figure 2). Our modification, which consists in braiding the knot K along fibers (see Figure 6), as we will see increases the complexity of incompressible surfaces with verticality defect greater than two. The modification involves a choice (which is not unique) and the goal of the modification along fibers of double points is to avoid handles attaching that reduce the complexity of vertical surfaces like in Figure 2.

Lemma 4.1 Let $S_1$ and $S_2$ be two vertical surfaces in $M_{\hat {K}}$ on which we attach a handle $H_{\alpha }$ to obtained a surface $S:=S_1\underset {H_{\alpha }}{\#}S_2$ .

If $w(\alpha )\geq 2$ , then there is a surface $S'$ homologous to S such that

$$ \begin{align*}-\chi(S')<-\chi(S).\end{align*} $$

Proof. There are two alternatives concerning the configuration of a handle depending on wether $\pi (H_{\alpha })$ contains a double point or not.

If $\pi (H_{\alpha })$ does not contain a double point of $\overset {\rightarrow }{\gamma }$ , then S is compressible. In fact, the curve $\beta $ (Figure 7-a) which is obtained by summing two fibers in $S_1$ and $S_2$ along $H_{\alpha }$ is essential in S (since fibers are essential in $S_1$ and $S_2$ ) and bounds a disk in $M_{\hat {K}}$ (see Figure 7a). So, we can reduce the complexity of S in this case by cutting S along the disk bounded by $\beta $ .

Figure 7 (a) Compression disk in $M_{\hat {K}}$ bounded by an essential curve $\beta $ in S. (b) The arc around the fiber of a double point which intersects a rectangle. This shows that the rectangle cannot go completely along the fiber.

Now, suppose that $\pi (H_{\alpha })$ contains double points of $\overset {\rightarrow }{\gamma }$ . We claim that $H_{\alpha }$ is horizontal. Since $w(\alpha )\geq 2$ , $H_{\alpha }$ is isotopic to a blow up of a rectangle between subarcs of $\hat {K}$ . A rectangle stays on one side of an arc. Thus, it cannot follow a subarc of $\hat {K}$ along a fiber (see Figure 7b).

Finally, if $H_{\alpha }$ is horizontal and $\pi (H_{\alpha })$ contains a double point p, then the fiber $\pi ^{-1}(p)$ intersects $H_{\alpha }$ twice. Therefore, $H_{\alpha }$ intersects $\hat {K}$ four times (at the braiding above p) and these intersection points define four boundary components on S. By attaching a new handle along the fiber $\pi ^{-1}(p)$ which encloses those four boundary components (see Figure 8) we obtain a surface $S'$ with one more handle and four boundary components less. So $-\chi (S')\leq -\chi (S)$ .▪

Figure 8 Replacing four boundary components by attaching a handle which enclosed the modification along a fiber.

Now, we are able to prove the main theorem.

Homologically nontrivial polytopes realized by our construction do not have fibered faces since a fibration of $M_{\hat {K}}$ by vertical surfaces would give a foliation on $\Sigma _g$ without singularities.

Our main theorem links the realization problems of intersection norms and Thurston norms. In [Reference Sane7], we showed that every polytope P in $\mathcal {P}_8$ : the set of non degenerate symmetric subpolytopes of $[-1,1]^4$ with eight vertices, is not the dual unit ball of an intersection norm.

By Gabai’s theorem which states that norm-minimizing surfaces are leaves of foliations without Reeb component, this question is somehow related to the studying of the topology of foliated (without Reeb component) 3-manifolds with pairs of pants or one-holed torus as leaves.