Note on primitive disk complexes

We will describe an orientation preserving homeomorphism of $(V,W;\Sigma)$ as an isotopy from the identity to the homeomorphism. Choose a primitive system $\mathcal{D}=\{D_{1},D_{2},\ldots,D_{g}\}$ of $V$ such that $D_{1}=D$, together with a dual system $\mathcal{D}^{\prime}=\{D^{\prime}_{1},D^{\prime}_{2},\ldots,D^{\prime}_{g}\}$ of $W$, and then assign symbols $x_{1},x_{2},\ldots,x_{g}$ to the circles $\partial D^{\prime}_{1},\partial D^{\prime}_{2},\ldots,\partial D^{\prime}_{g}$ respectively. Fix a base point inside $W$ and choose pairwise disjoint arcs $a_{i}$, for $i\in\{1,2,\ldots,g\}$, connecting the base point to a point in $\partial D_{i}$ respectively so that each $a_{i}$ is disjoint from $D^{\prime}_{1}\cup D^{\prime}_{2}\cup\cdots\cup D^{\prime}_{g}$. Then, with a choice of orientations of boundary circles of $D_{i}$ and of $D^{\prime}_{j}$, for $i,j\in\{1,2,\ldots,g\}$, the circles $\partial D_{1},\partial D_{2},\ldots,\partial D_{g}$ represent the generators $x_{1},x_{2},\ldots,x_{g}$ of $\pi_{1}(W)=\langle x_{1},x_{2},\ldots,x_{g}\rangle$ respectively. See Figure 1.

First we construct the homeomorphisms $\phi_{\alpha},\phi_{\beta},\phi_{\gamma}$, and $\phi_{\delta}$ of the splitting $(V,W;\Sigma)$ that induce the four elementary Nielsen transformations $\alpha,\beta,\gamma$, and $\delta$ respectively.

For the transformation $\alpha$, we have the homeomorphism $\phi_{\alpha}$ of $(V,W;\Sigma)$ described by the isotopy in Figure 2 (a). We drag a foot of the first handle along the second handle to make the circle $\phi_{\alpha}(\partial D_{1})$ determine the word $x_{1}x_{2}$ with respect to $\mathcal{D}^{\prime}$, while $\phi_{\alpha}(\partial D_{j})$ determines $x_{j}$ for each $j\in\{2,3,\ldots,g\}$. After the isotopy for $\phi_{\alpha}$, all the disks in $\mathcal{D}\cup\mathcal{D}^{\prime}$ except $D_{1}$ and $D^{\prime}_{2}$ remain unchanged as subsets.

For the transformation $\beta$, the homeomorphism $\phi_{\beta}$ is described as a $\pi$-rotation of the first handle as in Figure 2 (b). The circle $\phi_{\beta}(\partial D_{1})$ determines the word $x^{-1}_{1}$ with respect to $\mathcal{D}^{\prime}$, while $\phi_{\beta}(\partial D_{j})$ determines $x_{j}$ for $j\in\{2,3,\ldots,g\}$. After the isotopy for $\phi_{\beta}$, all the disks in $\mathcal{D}\cup\mathcal{D}^{\prime}$ remain unchanged as subsets.

For the transformation $\gamma$, the homeomorphism $\phi_{\gamma}$ described by the isotopy in Figure 2 (c) exchanges the first and the second handles so that $\phi_{\gamma}(\partial D_{1})$ and $\phi_{\gamma}(\partial D_{2})$ determine $x_{2}$ and $x_{1}$ respectively with respect to $\mathcal{D}^{\prime}$, while $\phi_{\gamma}(\partial D_{j})$ determines $x_{j}$ for $j\in\{3,4,\ldots,g\}$. The isotopy for $\phi_{\gamma}$ exchanges $D_{1}$ and $D^{\prime}_{1}$ with $D_{2}$ and $D^{\prime}_{2}$ respectively, but all other disks in $\mathcal{D}\cup\mathcal{D}^{\prime}$ remain unchanged.

Finally, the homeomorphism $\phi_{\delta}$ described by the isotopy in Figure 2 (d), which is a permutation of each handle to the next one, induces the transformation $\delta$. For each $j\in\{1,2,\ldots,g-1\}$, $\phi_{\delta}(\partial D_{j})$ determines $x_{j+1}$, and $\phi_{\delta}(\partial D_{g})$ determines $x_{1}$ with respect to $\mathcal{D}^{\prime}$. The isotopy for $\phi_{\delta}$ sends $D_{j}$ and $D^{\prime}_{j}$ to $D_{j+1}$ and $D^{\prime}_{j+1}$ respectively for $j\in\{1,2,\ldots,g-1\}$, and sends $D_{g}$ and $D^{\prime}_{g}$ to $D_{1}$ and $D^{\prime}_{1}$ respectively.

For each of the four homeomorphisms $\phi_{\alpha}$, $\phi_{\beta}$, $\phi_{\gamma}$, and $\phi_{\delta}$, we make the union of arcs $a_{1}\cup a_{2}\cup\cdots\cup a_{g}$ remain invariant after the isotopies. Furthermore we observe that, if $\phi$ is one of $\phi_{\alpha},\phi_{\beta},\phi_{\gamma}$, or $\phi_{\delta}$, then either $D_{j}=\phi(D_{j})$ or $D_{j}$ and $\phi(D_{j})$ are disjoint from each other for each $j\in\{1,2,\ldots,g\}$, and in this case we can easily find dual disks $E^{\prime}$ and $E^{\prime\prime}$ of $D_{j}$ and $\phi(D_{j})$ respectively, satisfying that $E^{\prime}$ is disjoint from $\phi(D_{j})\cup E^{\prime\prime}$ and $E^{\prime\prime}$ is disjoint from $D_{j}\cup E^{\prime}$. For example, for $D_{1}$ and $\phi_{\alpha}(D_{1})$, we can take $E^{\prime}=\phi_{\alpha}(D^{\prime}_{2})$ and $E^{\prime\prime}=D^{\prime}_{2}$.

Simply we say that the homeomorphisms $\phi_{\alpha}$, $\phi_{\beta}$, $\phi_{\gamma}$, and $\phi_{\delta}$ are realizations of $\alpha$, $\beta$, $\gamma$, and $\delta$ with respect to the dual pair $\{\mathcal{D},\mathcal{D}^{\prime}\}$.

Next, let $\eta$ be any automorphism of $\pi_{1}(W)$. Then $\eta$ can be written as a finite product of elementary Nielsen transformations. That is, $\eta=\eta_{n-1}\eta_{n-2}\cdots\eta_{1}$ for some $n\geq 2$ where $\eta_{i}$ is one of $\alpha$, $\beta$, $\gamma$, and $\delta$. For $\eta_{1}$, we find a realization $\phi_{1}$, which is one of $\phi_{\alpha}$, $\phi_{\beta}$, $\phi_{\gamma}$, or $\phi_{\delta}$ with respect to the dual pair $\{\mathcal{D},\mathcal{D}^{\prime}\}$, described in the above. For $\eta_{2}$, letting $\mathcal{D}_{1}=\{\phi_{1}(D_{1}),\phi_{1}(D_{2}),\ldots,\phi_{1}(D_{g})\}$ and $\mathcal{D}^{\prime}_{1}=\{\phi_{1}(D^{\prime}_{1}),\phi_{1}(D^{\prime}_{2}),\ldots,\phi_{1}(D^{\prime}_{g})\}$, we have a realization $\phi_{2}$, which is one of $\phi_{\alpha}$, $\phi_{\beta}$, $\phi_{\gamma}$, or $\phi_{\delta}$ with respect to the dual pair $\{\mathcal{D}_{1},\mathcal{D}^{\prime}_{1}\}$. Continuing the process, we have a realization $\phi_{i}$ of $\eta_{i}$ for each $i\in\{2,\ldots,n-1\}$, with respect to the dual pair $\{\mathcal{D}_{i-1},\mathcal{D}^{\prime}_{i-1}\}$, where $\mathcal{D}_{i-1}$ and $\mathcal{D}^{\prime}_{i-1}$ consist of disks $\phi_{i-1}\cdots\phi_{2}\phi_{1}(D_{j})$ and $\phi_{i-1}\cdots\phi_{2}\phi_{1}(D^{\prime}_{j})$ respectively, for $j\in\{1,2,\ldots,g\}$. Then the product $\phi_{n-1}\phi_{n-2}\cdots\phi_{1}$ is the desired homeomorphism $\phi_{\eta}$ that induces $\eta$, and letting $D=E_{1}$, and $\phi_{i-1}\cdots\phi_{2}\phi_{1}(D)=E_{i}$ for $i\in\{2,3,\ldots,n\}$ we have the desired sequence of primitive disks in $V$ satisfying the condition. ∎

Choose any orientation preserving homeomorphism $\phi$ of $(V,W;\Sigma)$ taking $D$ to $E$. Let $\eta$ be the automorphism of $\pi_{1}(W)$ induced by $\phi$. By Lemma 2.1, there exists another orientation preserving homeomorphism $\phi_{\eta}$ of $(V,W;\Sigma)$ that induces $\eta$, together with a sequence $E_{1},E_{2},\ldots,E_{n}$ of primitive disks in $V$ satisfying the condition in the theorem. It is clear that $E=\phi(D)$ and $E_{n}=\phi_{\eta}(D)$ are equivalent to each other. ∎