An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface

For $g\geq 1$ and $n\geq 0$, let $N_{g,n}$ denote a compact non-orientable surface of genus $g$ with $n$ boundary components, that is, $N_{g,n}$ is a surface obtained by removing $n$ open disks from a connected sum of $g$ real projective planes. For $g\geq 0$ and $n\geq 0$, let $\Sigma_{g,n}$ denote a compact orientable surface of genus $g$ with $n$ boundary components, that is, $\Sigma_{g,n}$ is a surface obtained by removing $n$ open disks from a connected sum of $g$ tori. As shown in Figure 1, we can regard $N_{g,n}$ as a surface obtained by attaching $g-2h$ Möbius bands to $g-2h$ boundary components of $\Sigma_{h,n+g-2h}$, for $\displaystyle 0\leq{h}<\frac{g}{2}$. We call these attached Möbius bands crosscaps.

The mapping class group of $N_{g,n}$, denoted by $\mathcal{M}(N_{g,n})$, is the group consisting of isotopy classes of all diffeomorphisms of $N_{g,n}$ which fix the boundary pointwise. The mapping class group of $\Sigma_{g,n}$, denoted by $\mathcal{M}(\Sigma_{g,n})$, is the group consisting of isotopy classes of all orientation-preserving diffeomorphisms of $\Sigma_{g,n}$ which fix the boundary pointwise. It is well known that $\mathcal{M}(\Sigma_{g,n})$ is generated by only Dehn twists (see [3, 17, 4]). On the other hand, $\mathcal{M}(N_{g,n})$ can not be generated by only Dehn twists. As generators of $\mathcal{M}(N_{g,n})$, other than Dehn twists, crosscap slides or crosscap transpositions are needed (see [16, 18]). Let us consider the subgroup of $\mathcal{M}(N_{g,n})$ generated by all Dehn twists, denoted by $\mathcal{T}(N_{g,n})$. We call $\mathcal{T}(N_{g,n})$ the twist subgroup of $\mathcal{M}(N_{g,n})$.

We now explain about the history of studies on presentations for $\mathcal{M}(\Sigma_{g,n})$, $\mathcal{M}(N_{g,n})$ and $\mathcal{T}(N_{g,n})$. Finite presentations for $\mathcal{M}(\Sigma_{g,n})$ were given by Hatcher-Thurston [10] and Harer [9], and subsequently simplified by Wajnryb [28] and Matsumoto [20] for $n\leq 1$. Gervais [8] and Labruère-Paris [15] gave finite presentations of $\mathcal{M}(\Sigma_{g,n})$ for $n\geq 2$. Gervais [7] gave an infinite presentation for $\mathcal{M}(\Sigma_{g,n})$ by using the presentation for $\mathcal{M}(\Sigma_{g,n})$ given in [9, 28], and then Luo [19] simplified its presentation. Finite presentations for $\mathcal{M}(N_{2,0})$, $\mathcal{M}(N_{2,1})$, $\mathcal{M}(N_{3,0})$ and $\mathcal{M}(N_{4,0})$ ware given by [16], [23], [2] and [27] respectively. Note that $\mathcal{M}(N_{1,0})$ and $\mathcal{M}(N_{1,1})$ are trivial (see [5]). Paris-Szepietowski [22] gave a finite presentation of $\mathcal{M}(N_{g,n})$ for $g+n>3$ with $n\leq 1$. Stukow [25] gave another finite presentation of $\mathcal{M}(N_{g,n})$ for $g+n>3$ with $n\leq 1$, applying Tietze transformations for the presentation of $\mathcal{M}(N_{g,n})$ given in [22]. The second author [21] gave an infinite presentation of $\mathcal{M}(N_{g,n})$ for $g\geq 1$ and $n\leq 1$, using the presentation of $\mathcal{M}(N_{g,n})$ given in [25], and then, following this work, the authors [12] gave an infinite presentation of $\mathcal{M}(N_{g,n})$ for $g\geq 1$ and $n\geq 2$. It is known that $\mathcal{T}(N_{g,n})$ is the index $2$ subgroup of $\mathcal{M}(N_{g,n})$ (see [18]). Stukow [26] gave a finite presentation of $\mathcal{T}(N_{g,n})$ for $g+n>3$ with $n\leq 1$, applying the Reidemeister-Schreier method for the presentation of $\mathcal{M}(N_{g,n})$ given in [25] (see Theorems 2.5 and 2.6).

In this paper, we give an infinite presentation of $\mathcal{T}(N_{g,n})$ for $g\geq 1$ and $n\geq 0$ (see Theorem 1.1), mainly using the presentation of $\mathcal{T}(N_{g,n})$ given in [26] and Birman exact sequences on mapping class groups of non-orientable surfaces.

Through this paper, the product $gf$ of mapping classes $f$ and $g$ means that we apply $f$ first and then $g$. Moreover we do not distinguish a loop from its isotopy class.

For a simple closed curve $c$ of $N_{g,n}$, a regular neighborhood of $c$ is either an annulus or a Möbius band. We call $c$ a two sided or a one sided simple closed curve respectively. For a two sided simple closed curve $c$, we can take two orientations $+_{c}$ and $-_{c}$ of a regular neighborhood of $c$. The right handed Dehn twist $t_{c;\theta}$ about a two sided simple closed curve $c$ with respect to $\theta\in\{+_{c},-_{c}\}$ is the isotopy class of the map described as shown in Figure 2. $t_{c;\theta}$ does not depend on a choice of a representative curve of the isotopy class of $c$ and its regular neighborhood. We remark that although the Dehn twist was defined for an oriented simple closed curve in [21, 12], in this paper we do not consider an orientation of a simple closed curve in the definition of the Dehn twist. We write $t_{c;\theta}=t_{c}$ if the orientation $\theta$ is given explicitly. That is, the direction of the twist is indicated by an arrow written beside $c$ as shown in Figure 2.

We denote by $f_{\ast}(\theta)$ the orientation of a regular neighborhood of $f(c)$ induced from $\theta\in\{+_{c},-_{c}\}$, for a two sided simple closed curve $c$ of $N_{g,n}$ and $f\in\mathcal{M}(N_{g,n})$. Let $c_{1},\dots,c_{k}$, $c_{0}$, $c_{0}^{\prime}$ and $d_{1},\dots,d_{7}$ be simple closed curves with arrows as shown in Figure 3. $\mathcal{M}(N_{g,n})$ admits following relations.

1. (1)

   $t_{c;\theta}=1$ if $c$ bounds a disk or a Möbius band.

2. (2)

   $t_{c;-_{c}}^{-1}=t_{c;+_{c}}$.

3. (3)

   $ft_{c;\theta}f^{-1}=t_{f(c);f_{\ast}(\theta)}$ for $f\in\mathcal{M}(N_{g,n})$.

4. (4)

   $(t_{c_{1}}t_{c_{2}}\cdots{}t_{c_{k}})^{k+1}=t_{c_{0}}t_{c_{0}^{\prime}}$ if $k$ is odd,
   $(t_{c_{1}}t_{c_{2}}\cdots{}t_{c_{k}})^{2k+2}=t_{c_{0}}$ if $k$ is even.

5. (5)

   $t_{d_{1}}t_{d_{2}}t_{d_{3}}=t_{d_{4}}t_{d_{5}}t_{d_{6}}t_{d_{7}}$.

We can check the relations (1) and (2) easily. The relations (3), (4) and (5) are called a conjugation relation, a $k$-chain relation, a lantern relation respectively. These are famous relations on mapping class groups. In the relation (3), if $f=t_{c^{\prime};\theta^{\prime}}$, $|c\cap{}c^{\prime}|=0$ or $1$, and the orientations $\theta$ and $\theta^{\prime}$ are compatible, then the relation can be rewritten as a commutativity relation $t_{c;\theta}t_{c^{\prime};\theta^{\prime}}=t_{c^{\prime};\theta^{\prime}}t_{c^{;}\theta}$ or a braid relation $t_{c;\theta}t_{c^{\prime};\theta^{\prime}}t_{c;\theta}=t_{c^{\prime};\theta^{\prime}}t_{c^{;}\theta}t_{c^{\prime};\theta^{\prime}}$ respectively.

Our main result is as follows.

Here is the outline of this paper. In Section 2, we explain what we need to prove our main result. In Section 3, we prove that the infinitely presented group with the presentation in Theorem 1.1 is isomorphic to $\mathcal{T}(N_{g,n})$ with a presentation which is already known, for $n\leq 1$. In Section 4, we complete the proof of Theorem 1.1, by induction on $n$.

We first define the capping map, the point pushing map and the forgetful map on mapping class groups of non-orientable surfaces. Take a point $\ast$ in the interior of $N_{g,n-1}$. Let $\mathcal{M}(N_{g,n-1},\ast)$ denote the group consisting of isotopy classes of all diffeomorphisms of $N_{g,n-1}$ which fix $\ast$ and the boundary pointwise. We can regard $N_{g,n}$ as a complement of an open disk neighborhood of $\ast$ in $N_{g,n-1}$. The natural embedding $N_{g,n}\hookrightarrow{}N_{g,n-1}$ induces the homomorphism

which is called the capping map. The point pushing map

is defined as follows. For any loop $x\in\pi_{1}(N_{g,n-1},\ast)$, $\mathcal{P}_{\ast}(x)$ is described by pushing $\ast$ once along $x$. The forgetful homomorphism

is defined naturally, that is, $\mathcal{F}(f)$ does not necessarily fix $\ast$.

Next, we consider exact sequences on mapping class groups of non-orientable surfaces. Let $\mathcal{M}^{+}(N_{g,n-1},\ast)$ denote the subgroup of $\mathcal{M}(N_{g,n-1},\ast)$ consisting of elements which preserve a local orientation of $\ast$, and $\pi_{1}^{+}(N_{g,n-1},\ast)$ the subgroup of $\pi_{1}(N_{g,n-1},\ast)$ generated by two sided simple loops. We have the exact sequences

The second sequence is called the Birman exact sequence, introduced by Birman [1]. Note that $\mathcal{P}_{\ast}$ is injective except for the case $(g,n)=(2,1)$. For details see [1, 14, 24, 6, 22].

Finally, we define the crosscap pushing map. Let $S$ be a surface obtained by shrinking some crosscap of $N_{g,n}$ into a point. We call this operation the blowdown with respect to the crosscap. Note that $S$ is diffeomorphic to either $N_{g-1,n}$ or $\Sigma_{\frac{g-1}{2},n}$ (see Figure 6). Let $\ast$ be the shrinked point of $S$. Conversely, we can obtain $N_{g,n}$ from $S$, and call this operation the blowup with respect to $\ast$. The crosscap pushing map

is defined as follows. For $x\in\pi_{1}(S,\ast)$, let $\tilde{x}$ be an oriented loop of $N_{g,n}$ induced from $x$ by the blowup with respect to $\ast$. $\mathcal{P}_{\otimes}(x)$ is described by pushing the crosscap, which is obtained by the blowup with respect to $\ast$, once along $\tilde{x}$ (see Figure 7).

Similar to Remarks 2.1 and 2.2, we have the followings.

For $g\geq 2$ and $n\geq 0$ we have the short exact sequence

(see [16, 18]). Using the Reidemeister Schreier method for a presentation of $\mathcal{M}(N_{g,n})$, we can obtain a presentation of $\mathcal{T}(N_{g,n})$.

For $n\leq 1$, let $\alpha_{1},\dots,\alpha_{g-1}$, $\beta$, $\gamma$, $\varepsilon$ and $\zeta$ be simple closed curves of $N_{g,n}$ with arrows as shown in Figure 8, and if $g$ is even, $\beta_{0},\dots,\beta_{\frac{g-2}{2}}$, $\bar{\beta}_{\frac{g-6}{2}}$, $\bar{\beta}_{\frac{g-4}{2}}$ and $\bar{\beta}_{\frac{g-2}{2}}$ simple closed curves of $N_{g,n}$ with arrows as shown in Figure 8. Let $\alpha$ be an oriented simple loop of $N_{g-1,n}$ based at $\ast$ induced from $\alpha_{1}$ by the blowdown with respect to the first crosscap, as shown in Figure 8. For simplicity, we denote $t_{\alpha_{i}}=a_{i}$, $t_{\beta}=b$, $t_{\gamma}=c$, $t_{\varepsilon}=e$, $t_{\zeta}=f$, $t_{\beta_{i}}=b_{i}$, $t_{\bar{\beta}_{i}}=\bar{b}_{i}$ and $\mathcal{P}_{\otimes}(\alpha)=y$. Note that $y^{2}=t_{\delta}$, where $\delta$ is a simple close curve of $N_{g,n}$ with an arrow as shown in Figure 8. For $g\geq 1$ and $n\leq 1$, $\mathcal{T}(N_{g,n})$ admits a finite presentation given in Theorems 2.5, 2.6 or Lemma 2.7.