Return to Question

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.

In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:

Let $\Sigma$ be an oriented compact surface. Let $C_1,\dots,C_n \subset \Sigma$ be noncontractible disjoint circles contained in $\Sigma$, and for every $i=1,\dots,n$ let $T_i$ be a positively oriented Dehn twist around $C_i$. Also, for every $i=1,\dots,n$ choose a sign $\sigma_i \in \{\pm 1\}$. Assume that if a component of $\Sigma - \bigcup C_i$ is a cylinder bounded by $C_i$ and $C_j$, then $\sigma_i = \sigma_j$. Let $T := T_1^{\sigma_1} \circ \dots \circ T_n^{\sigma_n}$.
If $\gamma$ is a path in $\Sigma$ whose endpoints are fixed by $T$, and $T \circ \gamma$ is homotopic to $\gamma$ relative to the endpoints, then the endpoints of $\gamma$ lie on the same connected component of $\mathrm{Fix } T$.

I guess this is an (almost) trivial fact for experts on MCG of surfaces, hence the lack of proof, but could someone provide me a reference where this is proven?

Remark. For his purposes, Seidel assumes that $\Sigma$ has no boundary and that its genus is more than 1, but I think the claim is still true in this generality.

Thank you in advance for your time.

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.

In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:

Let $\Sigma$ be an oriented compact surface. Let $C_1,\dots,C_n \subset \Sigma$ be noncontractible disjoint circles contained in $\Sigma$, and for every $i=1,\dots,n$ let $T_i$ be a positively oriented Dehn twist around $C_i$. Also, for every $i=1,\dots,n$ choose a sign $\sigma_i \in \{\pm 1\}$. Assume that if a component of $\Sigma - \bigcup C_i$ is a cylinder bounded by $C_i$ and $C_j$, then $\sigma_i = \sigma_j$. Let $T := T_1^{\sigma_1} \circ \dots \circ T_n^{\sigma_n}$.
If $\gamma$ is a path in $\Sigma$ whose endpoints are fixed by $T$, and $T \circ \gamma$ is homotopic to $\gamma$ relative to the endpoints, then $\gamma$ is homotopic relative to the endpoints to a path $\gamma'$ contained in $\mathrm{Fix} T$.

I guess this is an (almost) trivial fact for experts on MCG of surfaces, hence the lack of proof, but could someone provide me a reference where this is proven?

Remark. For his purposes, Seidel assumes that $\Sigma$ has no boundary and that its genus is greater than 1, but I think the claim is still true in this generality.

Thank you in advance for your time.

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.

In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:

Let $\Sigma$ be an oriented compact surface. Let $C_1,\dots,C_n \subset \Sigma$ be noncontractible disjoint circles contained in $\Sigma$, and for every $i=1,\dots,n$ let $T_i$ be a positively oriented Dehn twits around $C_i$. Also, for every $i=1,\dots,n$ choose a sign $\sigma_i \in \{\pm 1\}$. Assume that if a component of $\Sigma - \bigcup C_i$ is a cylinder bounded by $C_i$ and $C_j$, then $\sigma_i = \sigma_j$. Let $T := T_1^{\sigma_1} \circ \dots \circ T_n^{\sigma_n}$.
If $\gamma$ is a path in $\Sigma$ whose endpoints are fixed by $T$, and $T \circ \gamma$ is homotopic to $\gamma$ relative to the endpoints, then the endpoints of $\gamma$ lie on the same connected component of $\mathrm{Fix } T$.

I guess this is an (almost) trivial fact for experts on MCG of surfaces, hence the lack of proof, but could someone provide me a reference where this is proven?

Remark. For his purposes, Seidel assumes that $\Sigma$ has no boundary and that its genus is more than 1, but I think the claim is still true in this generality.

Thank you in advance for your time.

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.

In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:

Let $\Sigma$ be an oriented compact surface. Let $C_1,\dots,C_n \subset \Sigma$ be noncontractible disjoint circles contained in $\Sigma$, and for every $i=1,\dots,n$ let $T_i$ be a positively oriented Dehn twist around $C_i$. Also, for every $i=1,\dots,n$ choose a sign $\sigma_i \in \{\pm 1\}$. Assume that if a component of $\Sigma - \bigcup C_i$ is a cylinder bounded by $C_i$ and $C_j$, then $\sigma_i = \sigma_j$. Let $T := T_1^{\sigma_1} \circ \dots \circ T_n^{\sigma_n}$.
If $\gamma$ is a path in $\Sigma$ whose endpoints are fixed by $T$, and $T \circ \gamma$ is homotopic to $\gamma$ relative to the endpoints, then the endpoints of $\gamma$ lie on the same connected component of $\mathrm{Fix } T$.

I guess this is an (almost) trivial fact for experts on MCG of surfaces, hence the lack of proof, but could someone provide me a reference where this is proven?

Remark. For his purposes, Seidel assumes that $\Sigma$ has no boundary and that its genus is more than 1, but I think the claim is still true in this generality.

Thank you in advance for your time.