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It's a standard result that the fundamental group of a genus g surface has presentation

$$\pi_1(\Sigma_g)=\langle a_1,b_1 \ldots a_g,b_g | \prod_i [a_i,b_i] =1 \rangle.$$

Does anyone have a picture of say a genus 3 surface where these generating loops $a_i,b_i$ are explicitly drawn onto the surface?

For example I have see pictures like the one in this answer the fundamental group of a Riemann surface with n points removed or https://www.researchgate.net/figure/Fundamental-group-of-a-genus-3-surface_fig1_226810380 but those sets of 6 loops are different and it's not clear to me which one is correct for the above presentation, or which pairs are $a_1,b_1$ or $a_2,b_2$ or $a_3,b_3$.

Thanks in advance for any help.

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  • $\begingroup$ That genus 3 picture does, indeed, give your presentation, up to an appropriate bijection $\{a,b,c,d,e,f\} \leftrightarrow \{a_1,b_1,a_2,b_2,a_3,b_3\}$. $\endgroup$
    – Lee Mosher
    Feb 27, 2021 at 1:54
  • $\begingroup$ Mayyyybe this picture is a little more standard, but I don't know if the presentation is going to be any easier to read from it. $\endgroup$
    – Lee Mosher
    Feb 27, 2021 at 1:58
  • $\begingroup$ Thanks very much Lee, that picture is really helpful. $\endgroup$
    – Fromage
    Feb 28, 2021 at 11:01
  • $\begingroup$ I've turned my comments into an answer. $\endgroup$
    – Lee Mosher
    Mar 1, 2021 at 0:36

1 Answer 1

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That genus 3 picture does, indeed, give your presentation, up to an appropriate bijection $\{a,b,c,d,e,f\} \leftrightarrow \{a_1,b_1,a_2,b_2,a_3,b_3\}$.

Maybe this picture is a little more standard, but I don't know if the presentation is going to be any easier to read from it.

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