2. Preliminaries

3. Key propositions

In this section, we provide an outline of the following key propositions for the proof of Theorem 1.1.

We also list the lemmas required for the key propositions. All propositions and lemmas in this section, except for Lemma 3.6, are taken from [Reference Long, Margalit, Pham, Verberne and Yao5, Section 2], with the surfaces modified to be nonorientable.

A multicurve is a finite collection of pairwise disjoint essential simple closed curves in $N$ . Note that a curve belonging to a multicurve may be one-sided or two-sided. A multicurve is separating if its complement has more than one component. We say that two curves $a$ and $b$ lie in the same side of a separating multicurve $m$ if they are disjoint from $m$ and lie in the same complementary component. We remark that a separating multicurve $m$ should contain at least one two-sided curve.

Note that Lemma 3.3 states that a curve $c$ is separating if (and only if) $\alpha (c)$ is separating. Here, we review the following notions from graph theory:

• • A graph is a join if its vertices are divided into two or more nonempty sets such that each vertex in one of the sets is connected by an edge to every vertex in the other sets.

• • The link of a set $A$ of vertices in a graph is the subgraph spanned by the set of vertices that are not in $A$ and are connected by an edge to each vertex in $A$ .

For example, for $c,d_1,\dots, d_k \in \mathcal{C}^{\dagger }(F)$ , $c$ is a vertex of the link of $\{d_1,\dots, d_k\}$ in $\mathcal{C}^{\dagger }(F)$ if and only if $c$ and $d_i$ are disjoint curves in $F$ for each $i=1,\dots, k$ .

Lemma 3.3 can be proved as in [Reference Long, Margalit, Pham, Verberne and Yao5] in the following way. We can observe that a multicurve $m$ is separating if and only if the link of $m$ is a join; this shows the former claim. The latter claim follows from the fact that two curves lie in the same side of $m$ if and only if they belong to the same set in the partition of the link of $m$ as a join.

We define the hull of a collection of curves in a surface to be the union of the curves along with any embedded disks bounded by the curves.

Note that we assume that $X$ consists only of two-sided curves in Lemma 3.4. With this restriction, we can prove Lemma 3.4 in the same way as in [Reference Long, Margalit, Pham, Verberne and Yao5, Lemma 2.4]. We include a proof of Lemma 3.4 to explain why we added such a restriction.

Proof of Lemma 3.4. For the forward direction, suppose that $d$ is a vertex of $\mathcal{C}^{\dagger }(N)$ that lies in the hull of $X$ . We assume that there exists a vertex $e$ of $\mathcal{C}^{\dagger }(N)$ which intersects $d$ but is disjoint from each curve in $X$ . Then, $e$ should be contained in $N\setminus X$ . Since $d$ lies in the hull of $X$ and $e$ intersects with $d$ , the curve $e$ should be contained in a connected component of $N\setminus X$ which is a disk. Then it follows that $e$ is an inessential curve bounding a disk. This contradicts the assumption that $e$ is a vertex of $\mathcal{C}^{\dagger }(N)$ .

For the other direction, suppose that $d$ is a vertex of $\mathcal{C}^{\dagger }(N)$ that does not lie in the hull of $X$ (i.e., $d$ is not a vertex of $X$ nor a curve that lies in the union of disks bounding the curves in $X$ ). This means that there is a component of $d\setminus X$ which lies in a component $R$ of $N\setminus X$ that is not a disk. If the genus (as either orientable or nonorientable surface) of $R$ is at least $1$ , it is clear that there exists a curve in $R$ which is essential in $N$ . If the genus of $R$ is $0$ , then each curve in $R$ parallel to a boundary component of $R$ is an essential curve (here we use two-sidedness of the curves corresponding to the vertices of $X$ ). In particular, $R$ contains an essential curve in $N$ which intersects $d$ , as desired.

Proof. Assume that $c$ is one-sided. Then $c$ is nonseparating. We prove that $\alpha (c)$ is also one-sided. By Lemma 3.3, $\alpha (c)$ is also nonseparating. Let $F$ and $F'$ be surfaces obtained by cutting $N_{g}$ along $c$ and $\alpha (c)$ , respectively.

If $F$ is orientable, then $F$ is homeomorphic to $S^1_{\frac{{g}-1}{2}}$ . Assume that $\alpha (c)$ is two-sided. Then $F'$ is homeomorphic to $N^2_{{g}-2}$ since $g$ is odd. Thus, we can take a nonseparating multicurve $\{d_1,\cdots, d_{{g}-2}\}$ in $F'$ . This implies that $\{\alpha (c),d_1,\cdots, d_{{g} -2}\}$ is a nonseparating multicurve in $N_{g}$ . By Lemma 3.3, $\{c,\alpha ^{-1}(d_1),\cdots, \alpha ^{-1}(d_{{g}-2})\}$ is also a nonseparating multicurve in $N_{g}$ , and it implies that $F$ admits a nonseparating multicurve $\{\alpha ^{-1}(d_1),\cdots, \alpha ^{-1}(d_{{g}-2})\}$ consisting of $({g} -2)$ curves. Hence, we obtain $\frac{{g}-1}{2} \geq{g}-2$ , but this contradicts the assumption that ${g} \geq 4$ . Therefore, we see that $\alpha (c)$ is one-sided.

If $F$ is nonorientable, then $F$ is homeomorphic to $N_{{g}-1}^1$ . Thus, we can take a nonseparating multicurve $\{d_1,\cdots, d_{{g}-1}\}$ in $F$ . We can see that $F'$ admits a nonseparating multicurve consisting of $({g}-1)$ curves by the same argument as above. Hence, the genus of $F'$ is $({g}-1)$ , and therefore $\alpha (c)$ is one-sided by Lemma 2.1.

If $\alpha (c)$ is one-sided, then $c=\alpha ^{-1}(\alpha (c))$ is also one-sided. By contraposition, if $c$ is two-sided, then $\alpha (c)$ is also two-sided.

Proof. As in the case of orientable surfaces [Reference Long, Margalit, Pham, Verberne and Yao5], we proceed Lemma 3.7 in four steps:

1. 1. $\alpha$ preserves the union of torus pairs and pants pairs consisting of only nonseparating curves

2. 2. $\alpha$ preserves the set of torus pairs

3. 3. $\alpha$ preserves the degenerate torus pairs

4. 4. $\alpha$ preserves the set of torus triples

Step 1. Let $c$ and $d$ be nonseparating two-sided curves that intersect. It is enough to show that the following three conditions are equivalent:

1. (1) The pair $\{c,d\}$ is a torus pair or a pants pair.

2. (2) There exists at most one other vertex of $\mathcal{C}^{\dagger }(N)$ that lies in the hull of $\{c,d\}$ .

3. (3) There exists at most one vertex of $\mathcal{C}^{\dagger }(N)$ whose link contains the link of $\{c,d\}$ .

Items (2) and (3) are equivalent by Lemma 3.4. Item (1) implies (2) since if $\{c,d\}$ is a torus pair or a pants pair, then its hull is $c \cup d$ . Now we prove that (2) does not hold if (1) does not (this means that (2) implies (1)).

We modify the argument in [Reference Long, Margalit, Pham, Verberne and Yao5]. Assume that (1) does not hold (i.e., $\{c, d\}$ is neither a torus pair nor a pants pair). If $\{c, d\}$ is a bigon pair, then the hull of $\{c, d\}$ admits infinitely many vertices, which means that (2) does not hold. Thus, it is sufficient to consider the case where $\{c, d\}$ is not a bigon pair. Then, $c \cap d$ has two distinct connected components, say $a_1$ and $a_2$ . They divide $c$ into two curves $c_1$ and $c_2$ : $c \setminus (a_1 \sqcup a_2) = c_1 \sqcup c_2$ . Similarly, they also divide $d$ into $d_1$ and $d_2$ We can take $a_1$ and $a_2$ so that $c \cap d_1 = \emptyset$ (e.g., if we consider a moving point on $d$ starting from a component $a_1$ , we can take $a_2$ as the component where the point hits $c$ for the first time). We also take two distinct connected components $b_1$ and $b_2$ of $c \cap d$ with $a_1 \neq b_1$ (it could be $a_1 = b_2$ or $a_2 = b_1$ ). They divide $c$ into $c'_1,c'_2$ and divide $d$ into $d'_1,d'_2$ . We can take $b_1$ and $b_2$ to satisfy that $c'_1 \cap d = \emptyset$ and $c'_1 \neq c_1$ (we can choose $c'_1 = c_2$ if $a_1 = b_2$ and $a_2 = b_1$ ). In $c \cup d$ , there are four distinct simple closed curves $e_1$ , $e_2$ , $e_3$ , and $e_4$ that contain $c_1 \cup d_1$ , $c_2 \cup d_1$ , $c'_1 \cup d'_1$ , and $c'_2 \cup d'_1$ , respectively; they are all distinct from both $c$ and $d$ .

If we assume that (2) holds (for contradiction), then the following holds: at least one of the pairs $\{e_1,e_2\}$ and $\{e_3,e_4\}$ consists only of inessential curves. Let $e_1$ and $e_2$ be inessential without loss of generality. Then, we observe contradictions as follows:

• • If $e_1$ or $e_2$ bounds a disk, then the hull of $\{c,d\}$ contains infinitely many distinct curves; this contradicts the assumption that (2) holds.

• • If both $e_1$ and $e_2$ bound a Möbius band, then they represent the same homology class $[e_1]=[e_2]$ of $H_1(N,\mathbb{Z}/2\mathbb{Z})$ . Then $[c]=[e_1]+[e_2] = 0$ in $H_1(N,\mathbb{Z}/2\mathbb{Z})$ ; this means that $c$ is separating, which contradicts the assumption that $c$ is nonseparating.

Therefore, we have finished the proof of Step 1.

Step 2. By Lemma 3.3, it suffices to prove the following. Let $c$ and $d$ be nonseparating two-sided curves that form a torus pair or a pants pair. Then, the pair $\{c,d\}$ is a torus pair if and only if there exists a separating curve $e$ which is disjoint from both $c$ and $d$ , and all nonseparating simple closed curves in $N$ lying in the same side of $e$ as $\{c,d\}$ intersect $c\cup d$ . We now follow the proof of [Reference Long, Margalit, Pham, Verberne and Yao5, Step 2 in Lemma 2.5].

Let $\{c,d\}$ be a torus pair (then both $c$ and $d$ are nonseparating by definition). Let $R$ be a neighborhood of $c\cup d$ which is homotopic to a torus with one boundary component $e$ (here, we use the two-sidedness of torus pairs). The resulting surface by cutting $R$ along $c\cup d$ is an annulus. Any nonseparating curve in $R$ is not homotopic to $e$ . Hence, any nonseparating curve in $R$ intersects with $c\cup d$ . For the other direction, we assume that $c$ and $d$ are nonseparating two-sided curves in $N$ forming a pants pair $\{c,d\}$ . Let $e$ be any separating curve which is disjoint from $c\cup d$ , and let $R$ be the subsurface of $N$ that contains $c\cup d$ and has boundary $e$ . Then, $R$ needs to have a positive genus since $N$ is closed and $c$ and $d$ are not homotopic. There exists a closed neighborhood of $c\cup d$ which is a pair of pants $P$ contained in $R$ . We let $P^{\circ }$ denote the interior of $P$ . Since the genus of $P$ is $0$ , there exists a curve in $R\setminus P^{\circ }$ that is nonseparating in $R$ (and hence in $N$ ). This completes the proof of Step 2.

Step 3 and Step 4 can be proved as in [Reference Long, Margalit, Pham, Verberne and Yao5] as follows, and thus we obtain Lemma 3.7. We can observe that a torus pair $\{c,d\}$ is nondegenerate if and only if there is exactly one other vertex of $\mathcal{C}^{\dagger }(N)$ in the hull of $\{c,d\}$ ; Step 3 follows from this observation and Lemma 3.4. Step 4 follows from Step 2, Lemma 3.4 and the fact that if $\{c,d,e\}$ is a torus triple, then $e$ is the exactly one other vertex in the hull of $\{c,d\}$ .

We explain the definition of an annulus set. Suppose that $(a,b)$ is an ordered pair of vertices of $\mathcal{C}^{\dagger }(N)$ that are disjoint, homotopic two-sided curves. Assuming ${g} \geq 3$ , there is a unique annulus $A$ in $N_{{g}}$ whose boundary is $a\cup b$ . Let $\mathcal{C}^{\dagger }(a,b)$ be the set of vertices of $\mathcal{C}^{\dagger }(N_{{g}})$ consisting of curves contained in the interior of $A$ . We refer to $\mathcal{C}^{\dagger }(a,b)$ as an annulus set. We call a pair of vertices of $\mathcal{C}^{\dagger }(N_{g})$ is an annulus pair if they lie in some annulus set $\mathcal{C}^{\dagger }(a,b)$ . A nonseparating noncrossing annulus pair is an annulus pair where both curves are nonseparating and the curves are noncrossing. There is a natural partial ordering on the annulus set $\mathcal{C}^{\dagger }(a,b)$ : we say that $c\preceq d$ if $c$ and $d$ are noncrossing and each component of $c\setminus d$ lies in the component of $A\setminus d$ bounded by $a$ .

Note that if two curves $a$ and $b$ on $N$ are disjoint and homotopic, then it automatically follows that $a$ and $b$ are two-sided since any pair of homotopic one-sided curves must intersect.

Proof of Lemma 3.9. For two disjoint nonseparating two-sided curves $a,b$ such that $\{a,b\}$ is a separating multicurve, we can observe, by Lemma 2.2, that the following two conditions are equivalent:

• • At least one component of the surface obtained by cutting $N_{g}$ along $a$ and $b$ is homeomorphic to $S_0^2$ or $N_1^2$ .

• • All separating curves disjoint from $a$ and $b$ lie in the same side of $\{a,b\}$ (if they exist).

Let $F$ and $F'$ (resp. $\tilde{F}$ and $\tilde{F}'$ ) denote the connected components of the surface obtained from $N_{g}$ by cutting along $a$ and $b$ (resp. $\alpha (a)$ and $\alpha (b)$ ). Since $a$ and $b$ are homotopic, we can assume that $F \cong S_0^2$ and $F' \cong N_{{g}-2}^2$ . By Lemma 3.3 and the above observation, $\tilde{F} \sqcup \tilde{F}'$ is homeomorphic to $S_0^2 \sqcup N_{{g}-2}^2$ or $N_1^2\sqcup N_{{g}-3}^2$ . Now we assume that the latter holds (to derive a contradiction). We assume that $\tilde{F} \cong N_1^2$ . Take a one-sided curve $c$ lies in $\tilde{F}$ . Since $\alpha ^{-1}(c)$ is one-sided by Lemma 3.6, the curve $\alpha ^{-1}(c)$ must lie in $F'$ . If we take a separating essential curve $d$ that lies in $F'$ (it exists since ${g} \geq 4$ ), then $\alpha (d)$ is separating and lies in $\tilde{F}$ by Lemma 3.3. This contradicts Lemma 2.2 and hence (1) follows.

Item (2) follows from (1) and Lemma 3.3. In (3) and (4), we discuss in an annular domain, and it does not depend on the orientability of the surface. Therefore, they are shown by exactly the same arguments as in [Reference Long, Margalit, Pham, Verberne and Yao5], so we omit their proofs.

Now we define type 1 and type 2 curves. Suppose that $\{c,d\}$ is a nonseparating noncrossing annulus pair, and suppose that $e$ is a curve so that $\{c,e\}$ and $\{d, e\}$ are degenerate torus pairs. If $c\cap e$ and $d\cap e$ are the same point, then we say that $e$ is a type 1 curve for $\{c,d\}$ . Otherwise, we say that $e$ is a type 2 curve for $\{c,d\}$ .

We can also discuss Lemma 3.10 in an annular domain as well, so its proof is the same as in [Reference Long, Margalit, Pham, Verberne and Yao5, Lemma 2.7], and therefore we omit it.

We end this section by discussing how Propositions 3.1 and 3.2 can be derived from the lemmas in this section. For Proposition 3.1, each nonseparating bigon pair is a nonseparating noncrossing annulus pair $\{c,d\}$ that forms exactly one inessential curve bounding a disk and has the additional property that $c\cap d$ is a nondegenerate interval. Hence, by Lemmas 3.9 and 3.10, we can prove Proposition 3.1 by the same way as the case of orientable surfaces. Proposition 3.2 follows from Proposition 3.1, Lemma 3.7, and the following claim: two bigon pairs $\{c,d\}$ and $\{c',d'\}$ form a linked sharing pair if and only if the following conditions hold.

1. 1. Each of $\{c,d'\}$ and $\{c',d\}$ is a nondegenerate torus pair.

2. 2. There is a curve that forms a torus triple with both $\{c,d'\}$ and $\{c',d\}$ .

The proof of claim is the same as the case of orientable surfaces.

4. Connectedness of fine arc graphs

In [Reference Long, Margalit, Pham, Verberne and Yao5], Long, Margalit, Pham, Verberne, and Yao introduced several fine arc graphs of orientable surfaces and proved their connectedness. In this section, we consider the fine arc graphs of nonorientable surfaces and prove their connectedness as well. The proofs for nonorientable surfaces are similar to those in [Reference Long, Margalit, Pham, Verberne and Yao5], but we include them for the convenience of the reader. Throughout this section, let $F$ be a compact surface with nonempty boundary $\partial F \neq \emptyset$ .

We define the fine arc graph ${\mathcal{A}^{\dagger }} (F)$ of $F$ . An arc $a \colon [0,1] \to F$ is said to be simple if $a$ is injective, proper if $a^{-1}(\partial F)=\{0,1\}$ , and essential if it is not homotopic to $\partial F$ . We assume an arc to be simple, essential, and proper. We say that two arcs have disjoint interiors if they are disjoint away from $\partial F$ . The fine arc graph ${\mathcal{A}^{\dagger }} (F)$ is the graph whose vertices are essential simple proper arcs in $F$ and whose edges connect vertices with disjoint interiors. We say that two arcs are completely disjoint if they have no intersections (including at the boundary).

Proof. Since the arc graph $\mathcal{A}(N)$ is connected (see [Reference Kuno4, Corollary 1]), by considering the natural projection ${\mathcal{A}^{\dagger }}(N) \to \mathcal{A}(N)$ , it suffices to show the following: for every two isotopic essential simple proper arcs in $N$ there exists a path in ${\mathcal{A}^{\dagger }}(N)$ connecting them. Let $a,b \colon [0,1] \to N$ be two isotopic arcs and $H \colon [0,1] \times [0,1] \to N$ an isotopy from $a$ to $b$ . For every $c \in{\mathcal{A}^{\dagger }}(N)$ , we define an open set $I_c$ in $[0,1]$ by:

\begin{equation*} I_c\;:\!=\;\{ t \in [0, 1] \mid a_t \text {is completely disjoint from} c \}, \end{equation*}

where $a_t \;:\!=\; H(\cdot, t) \colon [0,1] \to N$ . We can find $c_0,c_1,\dots, c_k \in{\mathcal{A}^{\dagger }}(N)$ such that

• • $[0,1] = I_{c_0} \cup I_{c_1} \cup \cdots \cup I_{c_k}$ ,

• • $0 \in I_{c_0}$ and $1 \in I_{c_k}$ , and

• • $I_{c_i} \cap I_{c_j} \neq \emptyset$ if and only if $|i-j| =1$ .

By taking $t_i \in I_{c_{i-1}} \cap I_{c_{i}}$ for each $i=1,\cdots, k$ , we obtain a path

\begin{equation*} a_0=a, c_0, a_{t_1}, c_1, \dots, c_{k-1}, a_{t_k}, c_k, a_1=b \end{equation*}

in ${\mathcal{A}^{\dagger }}(N)$ between $a$ and $b$ .

We say that an arc in $F$ is nonseparating if its complement in $F$ is connected. The fine nonseparating arc graph $\mathcal{NA}^{\dagger } (F)$ is the subgraph of ${\mathcal{A}^{\dagger }}(F)$ spanned by the nonseparating arcs.

Proof. Let $a,b \in \mathcal{NA}^{\dagger }(N)$ . By Proposition 4.1, there exists a path

\begin{equation*}a=a_0, a_1,\dots, a_{k-1}, a_k=b\end{equation*}

in ${\mathcal{A}^{\dagger }}(N)$ between $a$ and $b$ . We will obtain a path in $\mathcal{NA}^{\dagger }(N)$ between $a$ and $b$ from the above path by removing or replacing $a_i$ as necessary. Assume that $a_i$ is separating for some $i$ . If $a_{i-1}$ and $a_{i+1}$ lie in the other side of $a_i$ , then we remove $a_i$ . Otherwise, they lie in the same side of $a_i$ . Let $R$ be the subsurface obtained by cutting $N$ along $a_i$ and not including $a_{i-1}$ and $a_{i+1}$ . Then we can find a nonseparating arc $a_i'$ in $N$ which lie in $R$ as follows. Since $a_i$ is separating, its endpoints lie in the same component $d$ of $\partial N$ . If $R$ is planar, then $R$ contains some other component $d'$ of $\partial N$ since $a_i$ is essential; we can take $a_i'$ to be any arc in $R$ connecting $d$ to $d'$ . If $R$ has positive genus, then we can take $a_i'$ to be any nonseparating arc in $R$ connecting $d$ to itself. Therefore, by applying this operation to each separating arc $a_i$ , we obtain a path in $\mathcal{NA}^{\dagger }(N)$ between $a$ and $b$ .

For a surface $F$ with positive genus and a connected component $d_0$ of $\partial F$ , we will define the fine linked arc graph $\mathcal{A}_{\textrm{Lk}}^{\dagger }(F,d_0)$ . Let $a$ and $b$ be two vertices of ${\mathcal{A}^{\dagger }}(F)$ whose interiors are disjoint arcs. We say that $a$ and $b$ are linked at $d_0$ if all four endpoints lying in $d_0$ and a curve parallel and sufficiently close to $d_0$ intersects the arcs alternately. We define $\mathcal{A}_{\textrm{Lk}}^{\dagger }(F,d_0)$ as the graph whose vertices are nonseparating simple proper arcs in $F$ with both endpoints lying in $d_0$ and whose edges connect arcs with disjoint interiors that are linked at $d_0$ .

Proof. Let $a,b \in \mathcal{A}_{\textrm{Lk}}^{\dagger }(N,d_0)$ . By Corollary 4.2, there exists a path

\begin{equation*}a = a_0, a_1, \dots, a_{k-1}, a_k = b\end{equation*}

in $\mathcal{NA}^{\dagger }(N)$ . We can assume that the endpoints of each $a_i$ lie in $d_0$ . We will obtain a path from $a$ to $b$ in $\mathcal{A}_{\textrm{Lk}}^{\dagger }(N,d_0)$ from the above path as follows: for each pair $a_i$ and $a_{i+1}$ that are not linked, we find an arc $b_i$ that is linked with both $a_i$ and $a_{i+1}$ , and insert $b_i$ between $a_i$ and $a_{i+1}$ .

Assume that $a_i$ and $a_{i+1}$ are not linked. Then, a tubular neighborhood $A$ of $d_0$ in $N$ is divided into four components $A_0$ , $A_1$ , $A_2$ , and $A_3$ by arcs $a_i$ and $a_{i+1}$ . Let $A_0$ and $A_2$ be those that are bounded by one arc of $a_i$ and one arc of $a_{i+1}$ . It is sufficient to see that $A_1$ and $A_3$ lie in the same component of $N \setminus (a_i \cup a_{i+1})$ ; we can take $b_i$ as an arc connecting $A_1$ and $A_3$ in $N \setminus (a_i \cup a_{i+1})$ . Note that this is obviously true if $N \setminus (a_i \cup a_{i+1})$ is connected.

Suppose $N \setminus (a_i \cup a_{i+1})$ is not connected. Then both $a_i$ and $a_{i+1}$ must pass through crosscaps an even number of times (if $a_i$ or $a_{i+1}$ passes through crosscaps an odd number of times, then $N \setminus (a_i \cup a_{i+1})$ would be connected). By cutting $N$ along $a_i \cup a_{i+1}$ , the boundary $d_0$ splits into three boundaries $d_1$ , $d_2$ , and $d_3$ so that $A_i$ is attached to $d_i$ for $i=1,2,3$ ( $A_0$ is attached to $d_2$ ). It follows that $d_1$ and $d_3$ (and hence $A_1$ and $A_3$ ) must lie in the same components; otherwise, it contradicts the assumption that $a_i$ and $a_{i+1}$ are nonseparating. This completes the proof.

5. Automorphisms of the extended fine curve graphs

The goal of this section is to provide an outline of the proof of the following theorem:

We define the extended fine curve graph $\mathcal{EC}^{\dagger }(N)$ of a closed nonorientable surface $N$ to be a graph whose vertices are all the essential simple closed curves in $N$ or the inessential simple closed curves that bound a disk in $N$ , and an edge is a pair of vertices which are disjoint in $N$ . We emphasize that in our argument the vertex set of the extended fine curve graph $\mathcal{EC}^{\dagger }(N)$ of any closed nonorientable surface $N$ does not contain any inessential curves bounding a Möbius band.

Following the method of [Reference Long, Margalit, Pham, Verberne and Yao5], we use convergent sequences described below to prove Theorem 5.1. We see that the lemmas and corollaries corresponding to [Reference Long, Margalit, Pham, Verberne and Yao5, Section 4] also hold for nonorientable surfaces. We will only describe why they are valid for nonorientable surfaces and omit their proofs.

Let $N$ be a closed nonorientable surface of genus $g$ . We say that a sequence of vertices $(c_{i})$ of $\mathcal{EC}^{\dagger }(N)$ converges to a point $x\in N$ if every neighborhood of $x$ contains all but finitely many of the corresponding curves to $c_{i}$ , and we write $\mathrm{lim}(c_{i})=x$ . If $(c_{i})$ is convergent, it must be that there exists $M\gt 0$ such that each $c_{i}$ with $i\gt M$ bounds a disk in $N$ .

Since [Reference Long, Margalit, Pham, Verberne and Yao5, Lemma 4.1] is proved by a local argument in a surface, the same argument works also for nonorientable surfaces.

Since the argument in [Reference Long, Margalit, Pham, Verberne and Yao5, Lemma 4.2] is compactness of surfaces, we can show the same result for nonorientable surfaces.

We say that two convergent sequences of vertices of $\mathcal{EC}^{\dagger }(N)$ are coincident if they converge to the same point of $N$ . The interleave of two sequences $(c_{i})$ and $(d_{i})$ is a sequence $c_{1}$ , $d_{1}$ , $c_{2}$ , $d_{2}$ , $\cdots$ . We have the following corollary of Lemma 5.3.

We say that a sequence $(c_{i}^{1})$ , $(c_{i}^{2})$ , $(c_{i}^{3})$ , $\cdots$ of convergent sequences of vertices of $\mathcal{EC}^{\dagger }(N)$ converges if the sequence of limit points $\textrm{lim}(c_{i}^{1})$ , $\textrm{lim}(c_{i}^{2})$ , $\textrm{lim}(c_{i}^{3})$ , $\cdots$ converges to a point $x\in N$ . In this case, we say that the sequence converges to $x$ .

We say that a vertex $c$ is a limit curve for a sequence $(c_{i})$ of vertices in $\mathcal{EC}^{\dagger }(N)$ if $\textrm{lim}(c_{i})\in c$ .

We provide an outline of the proof of Theorem 5.1. We can prove the injectivity of $\nu$ as follows.

Proof. We can prove the same as in [Reference Long, Margalit, Pham, Verberne and Yao5] as follows. Assume that $f \in \operatorname{Ker} \nu$ . For every $x \in N$ , there exist two curves $c,d \in \mathcal{EC}^{\dagger }(N)$ such that $c \cap d = \{x\}$ . Then, $f(x)=f(c \cap d)=c \cap d = x$ and this implies that $f$ is the identity.

Therefore, it is sufficient to prove that $\nu$ is surjective. As in [Reference Long, Margalit, Pham, Verberne and Yao5], we can define $\xi \colon \textrm{Aut}(\mathcal{EC}^{\dagger }(N)) \to \textrm{Homeo}(N)$ as follows: for $\alpha \in \textrm{Aut}(\mathcal{EC}^{\dagger }(N))$ , we define $\xi (\alpha ) \in \textrm{Homeo}(N)$ by $\xi (\alpha )(x)=\lim ( \alpha (c_i))$ , where $(c_i)$ is a convergent sequence of $x \in N$ .

As Claims 1 and 2 in the proof of [Reference Long, Margalit, Pham, Verberne and Yao5, Lemma 4.6], we can observe the following:

1. (a) If $\alpha \in \textrm{Aut}(\mathcal{EC}^{\dagger }(N))$ and $c \in \mathcal{EC}^{\dagger }(N)$ , then $\xi (\alpha )(c)=\alpha (c)$ .

2. (b) If $f \in \textrm{Homeo}(N)$ and $c \in \mathcal{EC}^{\dagger }(N)$ , then $\nu (f)(c)=f(c)$ .

We remark that (a) follows from Corollary 5.6 and (b) follow from the definition. By using (a) and (b), for every $\alpha \in \textrm{Aut}(\mathcal{EC}^{\dagger }(N))$ and every $c \in \mathcal{EC}^{\dagger }(N)$ , we have

\begin{equation*} \nu (\xi (\alpha ))(c)=\xi (\alpha )(c)=\alpha (c). \end{equation*}

This implies that $\nu (\xi (\alpha ))= \alpha$ and thus $\nu$ is surjective, and Theorem 5.1 holds.

6. Automorphisms of the fine curve graph

In this section, we prove Theorem 1.1. By the same argument as in the proof of Lemma 5.7, the following lemma holds.

Before proving Theorem 1.1, we require the following notion. Let $\Gamma$ be a graph and $\Delta \subset \Gamma$ a subgraph. A map $E\colon \mathrm{Aut}(\Delta )\rightarrow \mathrm{Aut}(\Gamma )$ is an extension map if for any $\varphi \in \mathrm{Aut}(\Delta )$ , $E(\varphi )(\Delta )=\Delta$ , and $E(\varphi )|_{\Delta }=\varphi$ .

We now prove Theorem 1.1, by the same strategy as the proof of [Reference Long, Margalit, Pham, Verberne and Yao5, Theorem 1.1].

Proof of Theorem 1.1. This proof has two steps. In the first step, we construct an extension homomorphism $\varepsilon \colon \mathrm{Aut}(\mathcal{C}^{\dagger }(N))\rightarrow \mathrm{Aut}(\mathcal{EC}^{\dagger }(N))$ , and in the second step, we complete the proof by using the extension homomorphism $\varepsilon$ .

Step 1. Let $\alpha \in \mathrm{Aut}(\mathcal{C}^{\dagger }(N))$ . We define $\hat{\alpha }\colon \mathcal{EC}^{\dagger }(N)\rightarrow \mathcal{EC}^{\dagger }(N)$ as follows. If $c$ is an essential curve in $N$ , that is, if $c\in \mathcal{C}^{\dagger }(N)$ , then $\hat{\alpha }(c)=\alpha (c)$ . If $e$ is an inessential curve which bounds a disk in $N$ , then we choose a nonseparating bigon pair $\{c,d\}$ that determines $e$ and define $\hat{\alpha }(e)$ to be an inessential curve bounding a disk determined by the nonseparating bigon pair $\{\alpha (c), \alpha (d)\}$ ; this correspondence makes sense because of Proposition 3.1.

First, we verify $\hat{\alpha }$ is well defined and an automorphism of $\mathcal{EC}^{\dagger }(N)$ . Suppose that $\{c',d'\}$ is another nonseparating bigon pair that determines $e$ . It follows from Corollary 4.3 that there exists a sequence of bigon pairs:

\begin{equation*} \{c,d\}=\{c_{0},d_{0}\}, \{c_{1}, d_{1}\},\cdots, \{c_{n}, d_{n}\}=\{c',d'\}, \end{equation*}

where each pair $\{\{c_{i},d_{i}\}, \{c_{i+1}, d_{i+1}\}\}$ ( $i=0,1,\cdots, n-1$ ) is a linked sharing pair for $e$ . It follows from Proposition 3.2 that $\{\alpha (c'), \alpha (d')\}$ also determines the inessential curve $\hat{\alpha }(e)$ bounding a disk, and we see that $\hat{\alpha }$ is well defined.

Next, we show that $\hat{\alpha }$ is an automorphism of $\mathcal{EC}^{\dagger }(N)$ . Since $\hat{\alpha }$ is a bijection on the set of vertices of $\mathcal{EC}^{\dagger }(N)$ it suffices to show that $\hat{\alpha }$ maps any edge to an edge.

Let $a_{1}$ and $a_{2}$ be two distinct essential curves. Then $\hat{\alpha }(a_{1})=\alpha (a_{1})$ and $\hat{\alpha }(a_{2})=\alpha (a_{2})$ , and so $\hat{\alpha }(a_{1})$ and $\hat{\alpha }(a_{2})$ form an edge if and only if $a_{1}$ and $a_{2}$ form an edge. For the case where an edge is spanned by an essential curve $c$ and an inessential curve $e$ bounding a disk, we claim that there exists a nonseparating bigon pair $\{d_{1}, d_{2}\}$ that determines $e$ and is disjoint from $c$ . In fact, if $c$ is a separating curve in $N$ , we set the two connected components $N'$ and $N''$ of $N\setminus c$ , and we suppose that $e$ lies in $N'$ . Since $c$ is essential and $N$ is a closed nonorientable surface, the genera of $N'$ and $N''$ are at least $2$ . Hence, we can take two-sided (note that bigon pairs are constructed by only two-sided curves) nonseparating curves $d_{1}$ and $d_{2}$ on the component $N'$ containing $e$ so that the pair $\{d_{1}, d_{2}\}$ is a bigon pair that determines $e$ . If $c$ is a nonseparating curve in $N$ , the surface $F$ obtained by cutting $N$ along $c$ is homeomorphic to one of $N_{{g}-1}^{1}$ , $N_{{g}-2}^{2}$ , $S_{\frac{{g}-1}{2}}^{1}$ , or $S_{\frac{{g}-2}{2}}^{2}$ (see Subsection 2.1). Since we assume that the genus of $N$ is at least $4$ , we can take two-sided nonseparating curves $d_{1}$ and $d_{2}$ on the surface $F$ so that the pair $\{d_{1}, d_{2}\}$ is a bigon pair that determines $e$ , as desired.

Figure 1. Bigon pair for disjoint inessential curves bounding a disk in the nested case (left) and the unnested case (right).

For the case where two inessential curves $e$ and $f$ that bound a disk form an edge, similar to the case of orientable surfaces we claim that two inessential curves $e$ and $f$ bounding a disk are disjoint if and only if the following holds up to relabeling $e$ and $f$ : for every nonseparating bigon pair $\{c,d\}$ that determines $e$ , there is a bigon pair $\{c',d'\}$ that determines $f$ and is disjoint from $e$ . The forward direction is proved by direct construction (see Figure 1), and note that since the genus of $N$ is at least $4$ , $N$ has a subsurface homeomorphic to an orientable subsurface of genus $1$ . For the other direction, we assume that two inessential curves $e$ and $f$ that bound a disk intersect. We choose an intersection point $x\in e\cap f$ . We can take a nonseparating bigon pair $\{c,d\}$ that determines $e$ where $x$ is one of the vertices of the bigon. Let $\{c',d'\}$ be any bigon pair that determines $f$ . Then $x\in f\subset c'\cup d'$ , and so $c'\cup d'$ intersects $c$ , as desired. By this claim, we see that $\hat{\alpha }$ preserves the set of edges spanned by two inessential curves bounding a disk. Therefore, we see that $\hat{\alpha }$ is an automorphism of $\mathcal{EC}^{\dagger }(N)$ , in particular $\hat{\alpha }$ is an automorphism of $\mathcal{C}^{\dagger }(N)$ .

By the definition of $\varepsilon \colon \mathrm{Aut}(\mathcal{C}^{\dagger }(N))\rightarrow \mathrm{Aut}(\mathcal{EC}^{\dagger }(N))$ given by $\varepsilon (\alpha )=\hat{\alpha }$ , we have the desired extension map.

Step 2. Recall that $\eta \colon \mathrm{Homeo}(N)\rightarrow \mathrm{Aut}(\mathcal{C}^{\dagger }(N))$ and $\nu \colon \mathrm{Homeo}(N)\rightarrow \mathrm{Aut}(\mathcal{EC}^{\dagger }(N))$ are the natural homomorphisms. By Theorem 5.1, we see that $\nu$ is an isomorphism. Let $\varepsilon$ be the extension homomorphism constructed in the first step.

As with (b) in the previous section, the following holds by definition:

1. (c) If $f \in \textrm{Homeo}(N)$ and $c \in \mathcal{C}^{\dagger }(N)$ , then $\eta (f)(c)=f(c)$ .

By (a) and (c), for every $\alpha \in \textrm{Aut}(\mathcal{C}^{\dagger }(N))$ and every $c \in \mathcal{C}^{\dagger }(N)$ , we have

\begin{equation*} \eta (\xi (\varepsilon (\alpha )))(c)=\xi (\varepsilon (\alpha ))(c)=\varepsilon (\alpha )(c)=\alpha (c). \end{equation*}

This implies that $\eta (\xi (\varepsilon (\alpha )))= \alpha$ and thus $\eta$ is surjective. Since $\eta$ is injective by Lemma 6.1, we have finished the proof.