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Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:

$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{n}~\middle|~\prod_{i = 1}^{n} \gamma_{i}\prod_{i = 1}^{g} [\alpha_{i},\beta_{i}]~\right\rangle.$$ Let $J$ be a hyperelliptic involution on $S_{g,n}$ as depicted in the figure below:

enter image description here

i.e. $J$ fixes $r\leq2g+2$ punctures and permutes $n-r$ punctures. Then $J$ will induce an automorphism $J_*$ of $\pi_1(S_{g,n})$ unique up to $\mathrm{Inn}(\pi_1(S_{g,n}))$. Then I have the following questions:

Q1. What is a smart choice(in the sense of Q2) of loops and base-point in the figure above satisfying the above-mentioned presentation?

Q2. How to compute $J_*(\alpha_i)$, $J_*(\beta_i)$, and $J_*(\gamma_j)$ in terms of $\alpha_i, \beta_i, \gamma_j$ for that choice?

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    $\begingroup$ @MikhailKatz: since the question says “unique up to [inner automorphisms]”, the choice of base point doesn’t matter. $\endgroup$
    – HJRW
    Commented Aug 10, 2023 at 11:42
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    $\begingroup$ @MikhailKatz: different choices of base point lead to automorphisms that differ by an inner automorphism. This is proved in the first few lectures of any undergraduate course in algebraic topology. $\endgroup$
    – HJRW
    Commented Aug 10, 2023 at 15:35
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    $\begingroup$ @MikhailKatz: I suggest you look at Farb and Margalit’s Primer, especially the section on the Dehn—Nielsen—Baer theorem. This is all quite standard. $\endgroup$
    – HJRW
    Commented Aug 13, 2023 at 2:09
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    $\begingroup$ @MikhailKatz - you are correct that the hyperelliptic involution $J$ does not act on the fundamental group - this is because it does not fix any points, so cannot fix a base-point. However, the mapping class of $J$ gives an outer automorphism of the fundamental group (say with basepoint in front of the dotted line, halfway in-between the first two "x"s). As HJRW points out, this is the content of the (easy direction) of the Dehn-Nielsen-Baer theorem: en.wikipedia.org/wiki/… $\endgroup$
    – Sam Nead
    Commented Aug 13, 2023 at 9:59
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    $\begingroup$ @MikhailKatz: There seem to be several points of confusion here. Most of all, it's a real shame that you have chosen to hassle a new poster who asked a perfectly well-posed question. Regarding the maths: First, the crosses in the question indicate punctures rather than marked points (note that the given presenttion presents a free group), so there are no fixed points. Second, the Dehn--Nielsen--Baer theorem does resolve the ambiguity you now you say you worry about -- which seems to be different from your initial criticism, because it does not just involve base points... $\endgroup$
    – HJRW
    Commented Aug 13, 2023 at 17:21

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Below is a cartoon showing one "solution" for $S_{0, 4}$. I'd encourage you to think about $S_{0, 6}$ next, then deal with $S_{1,4}$, then deal with $S_{2, 6}$, and then ponder the general case. Note that there is never a unique solution; this is because one must make a choice of a point-pushing map to "restore" the basepoint.

Cartoon showing the action of J (and a point push) on the generators of the fundamental group

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