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Definition: A marking on a sphere $S$ with $g$ handles ($g<\infty$) is an ordered collection $\Sigma_p=\{[\alpha_j],[\beta_j]\}_{j=1}^g$ of elements $[\alpha_j],[\beta_j]$ all of them in a same fundamental group $\pi_1(p,S),p\in S$ such that $\Sigma_p$ generates $\pi_1(p,S)$ and $\prod\limits_{j=1}^g[[\alpha_j],[\beta_j]]=1$, where $[[\alpha_j],[\beta_j]]$ denotes the comutator of $[\alpha_j]$ and $[\beta_j]$ (namely, $[\alpha_j][\beta_j][\alpha_j]^{-1}[\beta_j]^{-1}$).

Are the answers to the following questions "Yes"? (I wish they were, cause then I could use them in the proof of a theorem...):

Question 1: Given two markings $\Sigma_p=\{[\alpha_j],[\beta_j]\}_{j=1}^g$ and $\Sigma'_p=\{[\alpha_j],[\beta_j]\}_{j=1}^g$ on a sphere $S$ with $g$ handles does the equations $$F([\alpha_j]):=[\alpha'_j],\,F([\beta_j]):=[\beta'_j], \, j=1,\dots,g$$ (which we could denote as $F(\Sigma_p)=\Sigma'_p$) define an automorphism $F:\pi_1(p,S)\to \pi_1(p,S)$?

The idea of question 1 is to create an isomorphism using the "send-basis-to-basis" process, but the problem is that $\Sigma_p$ and $\Sigma_p'$ are not really basis, since an element $[\alpha]\in \pi_1(p,S)$ might have not a unique representation in terms of elements of $\Sigma_p$ (or $\Sigma'_p$).

Question 2: If the answer to question 1 is "Yes", is it true that $F\in \mathrm{Inn}(\pi_1(p,S))$? I mean, there exist a $[\gamma]\in \pi_1(p,S)$ such that $$F([\alpha])=[\gamma][\alpha][\gamma]^{-1},\,\forall [\alpha]\in \pi_1(p,S)?$$

Any proofs or counterexamples?

EDIT: I think I've made some progress. Such an $F$ as defined in question 1 is an isomorphism.

If the genus $g$ is zero (the sphere case), the problem of question 2 is trivial (since $\pi_1(p,S)=\{e\}$). In the case $g=1$ (torus) question 2 is false! In fact, let $\Sigma_p=\{[\alpha_1],[\beta_1]\,:\, [\alpha_1][\beta_1][\alpha_1]^{-1}[\beta_1]^{-1}=1\}$ be any marking with basepoint $p$ on $S$. So $\Sigma'_p=\{[\alpha'_1],[\beta'_1]\}$ with $[\alpha'_1]:=[\alpha_1]$ and $[\beta'_1]=[\alpha_1][\beta_1]$ is a marking on $S$ with basepoint $p$. In fact, $$\begin{array}{rcl}\Pi_{i=1}^1[[\alpha'_i],[\beta'_i]]&=&[\alpha'_1][\beta'_1][\alpha'_1]^{-1}[\beta'_1]^{-1}\\ &=&[\alpha_1][\alpha_1][\beta_1][\alpha_1]^{-1}[\beta_1]^{-1}[\alpha_1]^{-1}\\ &=&1, \end{array}$$ and $[\alpha_1]=[\alpha_1']$, $[\beta_1]=[\beta'_1][\alpha'_1]^{-1}$ shows that $\langle \Sigma'_p\rangle=\pi_1(p,S)$.

Although, there is no $[\gamma]\in \pi_1(p,S)$ such that $F([\alpha])=[\gamma][\alpha][\gamma]^{-1},\forall [\alpha]\in \pi_1(p,S)$ because, since $\pi_1(p,S)$ is abelian, $[\gamma][\alpha][\gamma]^{-1}=[\alpha],\forall [\alpha],[\gamma]\in \pi_1(p,S)$, but $F\neq Id:\pi_1(p,S)\to \pi_1(p,S)$, because $F([\beta_1])\neq [\beta_1]$. So...

New Question: Is there any chances of the answer to question 2 to be "Yes", for $g\geq 2$? (Since $\pi_1(p,S)$ is not abelian for in these cases...)

1 Answers 1

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Question 2 has negative answer. There is an important group describing this failure, called, the (extended) mapping class group (MCG) of $S$, it is isomorphic to $Out(\pi_1(S))=Aut(\pi_1(S))/Inn(\pi_1(S))$, where $Inn$ denotes the subgroup of inner automorphisms. See: https://en.wikipedia.org/wiki/Mapping_class_group_of_a_surface#The_Dehn.E2.80.93Nielsen.E2.80.93Baer_theorem and references therein. Examples of nontrivial elements of MCG abound, for instance, all Dehn twists are such.

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    I wished that $F\in \mathrm{Inn}(\pi_1(S))$ precisely because I wanted to use the Dehn-Nielsen Theorem! If I knew that $F\in \mathrm{Inn}(\pi_1(S))$, then I would also know that exists a homeomorphism $f\in \mathrm{Homeo}_0(S)$ such that $\nu([f])=[F]\in \mathrm{Out}(\pi_1(S))$, where $\nu: \mathrm{MCG}(S)\to \mathrm{Out}(\pi_1(S))$ is the Dehn-Nielsen Theorem isomorphism... Could you please explain a little bit more precisely why the very Dehn-Nielsen Theorem implies that question 2 has negative answer? Thanks in advance! – 2017-01-31
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    @AndersonFelipeViveiros: I am not sure what you know and what you do not know. For instance, do you know that MCG is a nontrivial group unless the surface has genus 0? Do you know that every self-homeomorphism of the surface fixing a basepoint induces an automorphism of the fundamental group? – 2017-01-31
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    Yeah, I know both of these results and the Dehn-Nielsen Theorem (although only the statement, I didn't see the proof). And I have an intuitive notion of what a Dehn-Twist is: "cut the handle, twist and then glue it again..." – 2017-01-31
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    I was trying to understand the proof that the Teichmüller space $T_g$ of closed marked Riemann surfaces of genus $g$ is in bijection with the Teichmüller space $T(R)$ of a closed Riemann surface $R$ of genus $g$, given by orientation-preserving diffeomorfisms... I'm following the proof of this in the Chapter 1 of Imayoshi & Taniguchi's "Introduction to Theichmüller Spaces". If you have access to it, I was able to understand that the map $\Phi_\Sigma$ is well defined and injective (using the Dehn-Nielsen Theorem). But I'm struggling to proof it is surjective (using the Dehn-Nielsen Theorem). – 2017-01-31
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    Ooops! I think I understood your counterexample now... Since $MCG(S)$ is nontrivial for $g\geq 2$, we can build up a counterexample choosing an element $[f]\notin \mathrm{Homeo}_0(S)$, riight? – 2017-01-31
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    PS: Is not $MCG(S^2)=\{[id],[\alpha]\}$, where $\alpha:S^2\to S^2$ is the antipodal map? – 2017-01-31
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    @AndersonFelipeViveiros It depends on your definition. Most people assume that mapping classes preserve orientation. In any case, I was talking about genus >1. – 2017-02-01
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    @AndersonFelipeViveiros Yes, you understood me correctly. – 2017-02-01
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    I think the book is just sloppy and does not take into account that marking has to preserve orientation. There is a way to say so algebraically by looking at the maps of generators, but it is a bit tricky. – 2017-02-01
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    You were very helpfull. Thank you! – 2017-02-01