Finally, let us consider the remaining...

Proof.

Again we will divide the proof in four steps.

Step 1: The base orbifold must be |$\ast 2222,2\!\ast\!22,22\ast$| or 22 ×.

If |$f:\mathbb{R}^3/\Gamma\to \mathcal{B}$| is a Seifert fibration for Γ a space group, by Proposition 5.4 we can assume that |$\mathcal{B}=\mathbb{R}^2/\Gamma_H^{[v]}$|⁠, where |$[v]$| is an invariant direction of Γ. Now, if |$\mathbb S^2/\rho(\Gamma)=222$| then |$\rho(\Gamma)$| can be chosen, up to conjugation by a linear map, to be the group

$$\rho(\Gamma)=\langle\begin{bmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{bmatrix}\rangle,$$

and the invariant directions are |$\{[e_1], [e_2],[e_3]\}\subseteq \mathbb{R} P^2$|⁠, where |$e_1,e_2,e_3$| are the standard basis of |$\mathbb{R}^3$|⁠. Therefore for |$v=e_1,e_2,e_3$| we have

$$\rho(\Gamma_H^{[v]})=\langle\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix},\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix},\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}\rangle\cong D_4.$$

This implies that |$\mathbb{R}^2/\Gamma_{[v]}$| is non-orientable and has at least one cone point or corner point and each cone or corner point has singularity index equal to 2. From Table 1, |$\mathbb{R}^2/\Gamma_H^{[v]}$| is diffeomorphic to one of the following orbifolds:

$$\ast 2222\quad 2\!\ast\!22\quad22\!\ast\quad22\times.$$

Step 2: All possible Seifert invariants for a fibration of |$\mathcal{O}$| are listed in the table.

As in the previous case, using Theorem 3.8 one can check that those listed in the statement are all the possible Seifert fibrations with base orbifold as in Step 1.

Step 3: Seifert orbifolds in the first three lines are orientation-preserving diffeomorphic.

We will first show the following three equivalences

$$(2_02_0\ast_0)\cong(\ast_02_12_12_12_1)\qquad(2_02_1\ast_1)\cong(2_1\ast_02_12_1)\qquad (2_12_1\ast_0)\cong(2_02_0\overline\times)$$

by constructing, for each pair, a space group Γ that admits two invariant directions inducing the correct fibrations. In the next step we will prove by a similar method the existence of a diffeomorphism between |$(\ast_02_02_02_12_1)$| and |$(2_0\ast_02_02_0)$|⁠.

The main idea to construct the space group Γ is to apply concretely the construction in Corollary 4.7. Let

$$G=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & -1\\\end{bmatrix}, \begin{bmatrix} 0\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 1\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right)\rangle\lt \mathrm{Isom}(\mathbb{R}^2)$$

be the wallpaper group generated by two rotations of π with fixed points |$(0,1/4)$| and |$(1/2,1/4)$| and a reflection in the line y = 0. Figure 14 shows a fundamental domain of this group action, and |$\mathbb{R}^2/G$| is diffeomorphic to |$22\ast$|⁠. For fixed |$a,b\in\{0,1\}$|⁠, we can apply the construction in Corollary 4.7 to the fibration |$(2_a2_b\ast)$|⁠, where the boundary invariant is determined by the first two. This gives the space group:

$$\Gamma=\langle\left(A,\begin{bmatrix} 0\\1/2\\a/2\end{bmatrix}\right), \left(A,\begin{bmatrix}1\\1/2\\b/2\end{bmatrix}\right),\left(B,\begin{bmatrix}0\\0\\0\end{bmatrix}\right), \left(\mathrm{Id},\begin{bmatrix}0\\0\\1\end{bmatrix}\right)\rangle\lt \mathrm{Isom}(\mathbb{R}^2)\times \mathrm{Isom}(\mathbb{R})\ ,$$

where

$$A=\begin{bmatrix} -1 & 0&0\\ 0 & -1&0\\ 0&0&1\end{bmatrix}\qquad B=\begin{bmatrix} 1 & 0&0\\ 0 & -1&0\\ 0&0&-1\end{bmatrix}\ .$$

By construction Γ admits a vertical translation, |$[e_3]$| is an invariant direction, |$G=\Gamma_H^{[e_3]}$| and the fibration |$f_3:\mathbb{R}^3/\Gamma\to \mathbb{R}^2/G$| induced by the partition in lines parallel to |$[e_3]$| is equivalent to the fibration |$(2_a2_b\ast)$|⁠.

$On the left a fundamental domain for the action is shown, and on the right the quotient $ 22\ast$ is shown.$

Figure 14.

On the left a fundamental domain for the action is shown, and on the right the quotient |$ 22\ast$| is shown.

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It is easy to see that G admits two translation in the directions |$[e_1],[e_2]\in\mathbb{R} P^1$| and that these directions are preserved by G. By Proposition 5.9, Γ admits three linearly independent translations with directions preserved by Γ. Since Γ preserves only the directions |$[e_1],[e_2],[e_3]$|⁠, we have that |$\mathbb{R}^3/\Gamma$| has three Seifert fibrations |$f_1,f_2,f_3$| induced by the partition in lines with directions |$[e_1],[e_2]$| and |$[e_3]$|⁠. Furthermore |$f_i:\mathbb{R}^3/\Gamma\to\mathbb{R}^2/\Gamma_H^{[e_i]}$| has base orbifold |$\mathbb{R}^2/\Gamma_H^{[e_i]}$|⁠, where

$$\begin{aligned} \Gamma_H^{[e_1]}&=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 1/2\\a/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 1/2\\b/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle \\ \Gamma_H^{[e_2]}&=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\a/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 1\\b/2\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle. \end{aligned}$$

Let us now distinguish three cases. If a = 0 and b = 0, the group

$$\begin{aligned} \Gamma_H^{[e_2]}&=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 1\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle \\ &=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 1\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle \end{aligned}$$

is generated by the reflections in the edges of the square whose vertices have coordinates 0 or |$1/2$|⁠. Then |$\mathbb{R}^2/\Gamma_H^{[e_2]}$| is diffeomorphic to |$\ast2222$|⁠. Since |$(2_02_0\ast_0)$| does not have vertex points, then the Seifert invariants of the fibration f₂ over every corner point cannot be equal to |$0/2$| and are therefore equal to |$1/2$|⁠. Therefore |$\mathbb{R}^3/\Gamma$| is diffeomorphic to |$(\ast_02_12_12_12_1)$|⁠, where the boundary invariant can be determined by Theorem 3.8 (see Remark 3.13). The two inequivalent fibrations are shown in Figure 1.

If a = 0 and b = 1, the group

$$\begin{aligned} \Gamma_H^{[e_2]}&=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 1\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle\\ &=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 1\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle \end{aligned}$$

is generated by three reflections and a rotation of order 2 with fixed point |$(1/2,1/4)$|⁠.

$On the left a fundamental domain for the action is shown, and on the right the quotient $ 2\!\ast\!22$ is shown.$

Figure 15.

On the left a fundamental domain for the action is shown, and on the right the quotient |$ 2\!\ast\!22$| is shown.

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A fundamental domain is shown in Figure 15, and |$\mathbb{R}^2/\Gamma_{[e_2]}$| is diffeomorphic to |$ 2\!\ast\!22$|⁠. As in the previous case, since |$(2_02_1\ast_1)$| does not have vertex points, all local invariants of the fibration f₂ over a corner point must be |$1/2$|⁠. Moreover the singular locus of |$(2_02_1\ast_1)$| has two connected components, which implies that the local invariant for f₂ over the cone point must be |$1/2$|⁠. Indeed, if it was |$0/2$|⁠, then the fibration |$(2_0\ast_12_12_1)$| would have three connected components, see Figure 18 (E). Therefore |$\mathbb{R}^3/\Gamma$| is diffeomorphic to |$(2_1\ast_02_12_1)$|⁠, where the boundary invariant is determined by Remark 3.13 as before.

Finally, if a = 1 and b = 1, the group

$$\begin{aligned} \Gamma_H^{[e_1]}&=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 1/2\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle\\ &=\langle\left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 1/2\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle \end{aligned}$$

is generated by a vertical glide reflection and two order 2 rotations with fixed points (0, 0) and |$(0,1/2)$|⁠.

Figure 16.

On the left a fundamental domain for the action is shown, and on the right the quotient 22 × is shown.

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Figure 16 shows a fundamental domain, and |$\mathbb{R}^2/\Gamma_H^{[e_1]}$| is diffeomorphic to 22 ×. Since the singular locus |$(2_12_1\ast_0)$| has two connected components, the local invariant over a cone point for the fibration f₁ is necessarily |$0/2$|⁠. Therefore |$\mathbb{R}^3/\Gamma$| is diffeomorphic to |$(2_02_0\overline\times)$|⁠.

Step 4: The orbifolds |$(\ast_12_02_02_12_1)$| and |$(2_0\ast_02_02_0)$| are orientation-preserving diffeomorphic.

Consider now the wallpaper group

$$G=\langle \left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right), \left(\begin{bmatrix} -1 & 0\\ 0 & 1\\\end{bmatrix},\begin{bmatrix} 1\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\1\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right)\rangle$$

generated by the reflections in the edges of the square whose vertices have coordinates 0 or |$1/2$|⁠, so that |$\mathbb{R}^2/G$| is diffeomorphic to |$\ast2222$|⁠. As in the previous step, by applying the construction of Corollary 4.7 to the fibration |$(\ast_12_02_02_12_1)$|⁠, we define the space group:

$$\Gamma=\langle\left(A,\begin{bmatrix} 0\\0\\0\end{bmatrix}\right), \left(B,\begin{bmatrix}1\\0\\1/2\end{bmatrix}\right),\left(A,\begin{bmatrix}0\\1\\0\end{bmatrix}\right),\left(B,\begin{bmatrix}0\\0\\0\end{bmatrix}\right), \left(\mathrm{Id},\begin{bmatrix}0\\0\\1\end{bmatrix}\right)\rangle,$$

where now

$$A=\begin{bmatrix} 1 & 0&0\\ 0 & -1&0\\ 0&0&-1\end{bmatrix}\qquad B=\begin{bmatrix} -1 & 0&0\\ 0 & 1&0\\ 0&0&-1\end{bmatrix}\ .$$

By construction, Γ has a translation in the invariant direction |$[e_3]$|⁠, |$G=\Gamma_H^{[e_3]}$| and the fibration |$f_3:\mathbb{R}^3/\Gamma\to\mathbb{R}^2/\Gamma_H^{[e_3]}$| induced by the parallel lines with the direction |$[e_3]$| is equivalent to the fibration |$(\ast_12_02_02_12_1)$|⁠.

Now, observe that |$[e_2]$| is an invariant direction for Γ and that Γ contains the translation in the direction e₂, obtained by composing the first and third generators. Hence we consider the fibration

$$f_2:\mathbb{R}^3/\Gamma\to\mathbb{R}^2/\Gamma_H^{[e_2]}.$$

By a direct computation, we obtain

$$\begin{aligned} \Gamma_H^{[e_2]}&=\langle\left(\begin{bmatrix} 1 & 0\\ 0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 1\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\mathrm{Id},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle\\ &=\langle\left(\begin{bmatrix} 1 & 0\\ 0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 1\\1/2\end{bmatrix}\right),\left(\begin{bmatrix}-1 & 0\\0 & 1\\\end{bmatrix},\begin{bmatrix} 0\\0\end{bmatrix}\right),\left(\begin{bmatrix}1 & 0\\0 & -1\\\end{bmatrix},\begin{bmatrix} 0\\1\end{bmatrix}\right)\rangle\ . \end{aligned}$$

This group is generated by three reflections and an order 2 rotation with fixed point |$(1/2,1/4)$|⁠, again as shown in Figure 15. The quotient |$\mathbb{R}^2/\Gamma_H^{[e_2]}$| is therefore diffeomorphic to |$2\ast22$|⁠.

Since |$(\ast_12_02_02_12_1)$| has four vertex points (see Figure 17), the local invariant of both corner points for the fibration f₂ has to be equal to |$0/2$|⁠. By Theorem 3.8 the only possibilities for the fibration f₂ are |$(2_0\ast_02_02_0)$| and |$(2_1\ast_12_02_0)$|⁠. However we can exclude the latter since the singular locus of |$(\ast_12_02_02_12_1)$| has two connected components (Figure 17 again), while |$(2_1\ast_12_02_0)$| has connected singular locus, see Figure 18 (D). This shows that |$\mathbb{R}^3/\Gamma$| is orientation-preserving diffeomorphic to |$(\ast_12_02_02_12_1)$| and |$(2_0\ast_02_02_0)$|⁠.

$The singular locus of $(\ast_12_02_02_12_1)$, which has S3 as underlying manifold, and all the edges have the singularity index equal to 2.$

Figure 17.

The singular locus of |$(\ast_12_02_02_12_1)$|⁠, which has S³ as underlying manifold, and all the edges have the singularity index equal to 2.

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Step 5: The orbifolds in different lines are not diffeomorphic.

To distinguish the orbifolds in different lines of the table, we analyse their singular loci, which are easily found following Section 3.2. We summarize the structure of the singular loci in Table 2; see also Figure 18.

Table 2.

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Structure of the singular loci

Orbifold \|$\mathcal{O}$\|	Description of \|$\Sigma_{\mathcal{O}}$\|	Components of \|$\Sigma_{\mathcal{O}}$\|	Vertex points
\|$(2_02_0\ast_0)$\|	Four circles with singularity index= 2	4	0
\|$(2_02_1\ast_1)$\|	Two circles with singularity index= 2	2	0
\|$(2_12_1\ast_0)$\|	Two circles with singularity index= 2	2	0
\|$(2_0\ast_02_02_0)$\|	See Figure 18A	2	4
\|$(\ast_02_02_02_02_0)$\|	See Figure 18B	1	8
\|$(\ast_12_02_12_02_1)$\|	See Figure 18C	2	4
\|$(2_1\ast_12_02_0)$\|	See Figure 18D	1	4
\|$(2_0\ast_12_12_1)$\|	See Figure 18E	3	0
\|$(2_12_1\overline\times)$\|	Empty	0	0

Orbifold \|$\mathcal{O}$\|	Description of \|$\Sigma_{\mathcal{O}}$\|	Components of \|$\Sigma_{\mathcal{O}}$\|	Vertex points
\|$(2_02_0\ast_0)$\|	Four circles with singularity index= 2	4	0
\|$(2_02_1\ast_1)$\|	Two circles with singularity index= 2	2	0
\|$(2_12_1\ast_0)$\|	Two circles with singularity index= 2	2	0
\|$(2_0\ast_02_02_0)$\|	See Figure 18A	2	4
\|$(\ast_02_02_02_02_0)$\|	See Figure 18B	1	8
\|$(\ast_12_02_12_02_1)$\|	See Figure 18C	2	4
\|$(2_1\ast_12_02_0)$\|	See Figure 18D	1	4
\|$(2_0\ast_12_12_1)$\|	See Figure 18E	3	0
\|$(2_12_1\overline\times)$\|	Empty	0	0

Table 2.

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Structure of the singular loci

Orbifold \|$\mathcal{O}$\|	Description of \|$\Sigma_{\mathcal{O}}$\|	Components of \|$\Sigma_{\mathcal{O}}$\|	Vertex points
\|$(2_02_0\ast_0)$\|	Four circles with singularity index= 2	4	0
\|$(2_02_1\ast_1)$\|	Two circles with singularity index= 2	2	0
\|$(2_12_1\ast_0)$\|	Two circles with singularity index= 2	2	0
\|$(2_0\ast_02_02_0)$\|	See Figure 18A	2	4
\|$(\ast_02_02_02_02_0)$\|	See Figure 18B	1	8
\|$(\ast_12_02_12_02_1)$\|	See Figure 18C	2	4
\|$(2_1\ast_12_02_0)$\|	See Figure 18D	1	4
\|$(2_0\ast_12_12_1)$\|	See Figure 18E	3	0
\|$(2_12_1\overline\times)$\|	Empty	0	0

Orbifold \|$\mathcal{O}$\|	Description of \|$\Sigma_{\mathcal{O}}$\|	Components of \|$\Sigma_{\mathcal{O}}$\|	Vertex points
\|$(2_02_0\ast_0)$\|	Four circles with singularity index= 2	4	0
\|$(2_02_1\ast_1)$\|	Two circles with singularity index= 2	2	0
\|$(2_12_1\ast_0)$\|	Two circles with singularity index= 2	2	0
\|$(2_0\ast_02_02_0)$\|	See Figure 18A	2	4
\|$(\ast_02_02_02_02_0)$\|	See Figure 18B	1	8
\|$(\ast_12_02_12_02_1)$\|	See Figure 18C	2	4
\|$(2_1\ast_12_02_0)$\|	See Figure 18D	1	4
\|$(2_0\ast_12_12_1)$\|	See Figure 18E	3	0
\|$(2_12_1\overline\times)$\|	Empty	0	0

$The orbifolds in (A)–(C) and (E) have S3 as underlying manifold; the one in (D) has underlying manifold $\mathbb R P^3$. All the edges have the singularity index equal to 2. In (E) we can recognize the Borromean link, see also [18, Example 13.1.5] for a nice description of this orbifold. The figures (A) (B) (C) (D) (E) represent the singular set of the orbifolds $(2_0\ast_02_02_0)$ $(\ast_02_02_02_02_0)$ $(\ast_12_02_12_02_1)$ $(2_1\ast_12_02_0)$ and $(2_0\ast_12_12_1)$, respectively$

Figure 18.

The orbifolds in (A)–(C) and (E) have S³ as underlying manifold; the one in (D) has underlying manifold |$\mathbb R P^3$|⁠. All the edges have the singularity index equal to 2. In (E) we can recognize the Borromean link, see also [18, Example 13.1.5] for a nice description of this orbifold. The figures (A) (B) (C) (D) (E) represent the singular set of the orbifolds |$(2_0\ast_02_02_0)$| |$(\ast_02_02_02_02_0)$| |$(\ast_12_02_12_02_1)$| |$(2_1\ast_12_02_0)$| and |$(2_0\ast_12_12_1)$|⁠, respectively

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All the orbifolds in the first column of Table 2 can be distinguished using the information from the last two columns, with two exceptions.

|$(2_0\ast_02_02_0)$| and |$(\ast_12_02_12_02_1)$| have the same number of vertex points and components of the singular locus, but they are not diffeomorphic since in the former all vertex points lie in one connected component of the singular locus, while for the latter they are in different components. See Figure 18(A) and (C).
|$(2_02_1\ast_1)$| and |$(2_12_1\ast_0)$| have the same number of vertex points and components of the singular locus. Suppose by contradiction that they are orientation-preserving diffeomorphic. Since each of them has two inequivalent fibrations, there would exist a flat orbifold with four inequivalent fibrations with point orbifold equal to 222. This is impossible by Proposition 5.4 and Step 1, since such an orbifold can have at most three inequivalent fibrations, one for each invariant direction.

\|$(2_02_0\ast_0)$\|	\|$(\ast_02_12_12_12_1)$\|
\|$(2_02_1\ast_1)$\|	\|$(2_1\ast_02_12_1)$\|
\|$(2_12_1\ast_0)$\|	\|$(2_02_0\overline\times)$\|
\|$(2_0\ast_02_02_0)$\|	\|$(\ast_12_02_02_12_1)$\|
\|$(\ast_02_02_02_02_0)$\|
\|$(\ast_12_02_12_02_1)$\|
\|$(2_1\ast_12_02_0)$\|
\|$(2_0\ast_12_12_1)$\|
\|$(2_12_1\overline\times)$\|

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