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Let $S_{g,n}$ be a Riemann surface of genus $g$, with $n$ points removed. The mapping class group of $S_{g,n}$ is denoted by $\Gamma_{g,n}$.

Is there a reference where the abelianization of $\Gamma_{g,n}$ calculated (or at least for $g$ sufficiently large, are they trivial)?

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The following statement can be found in Section 5 of Low-dimensional homology groups of mapping class groups: a survey:

Theorem: Let $g \geq 1$. Then $$H_1(\Gamma_{g,r}^n,\mathbb{Z}) \simeq \left\{ \begin{array}{cl} \mathbb{Z}_{12} & \text{if $(g,r)=(1,0)$} \\ \mathbb{Z}^r & \text{if $g=1,r \geq 1$} \\ \mathbb{Z}_{10} & \text{if $g=2$} \\ 0 & \text{if $g \geq 3$} \end{array} \right.$$

$\Gamma^n_{g,r}$ denotes the mapping class group of a connected orientable surface of genus $g$ with $r$ boundary components and $n$ punctures. Precise references are given in the survey.

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    $\begingroup$ The survey writes "is isomorphic". Are these isomorphisms canonical enough to be called equalities? At least not in the case $(g,r)=(1,0)$, where the $\mathrm{PGL}_2(\mathbf{Z})$ (the genuine mapping class group!) action on the abelianization of MCG (the oriented one), cyclic of order 12, is not trivial (det $-1$ elements act by multiplication by $-1$). $\endgroup$
    – YCor
    Oct 27, 2019 at 12:55
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    $\begingroup$ I replaced $=$ with $\simeq$ to avoid any confusion. $\endgroup$
    – AGenevois
    Oct 27, 2019 at 17:12
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    $\begingroup$ @YCor the outer automorphism (reversing orientation) acts on all these groups by $-1$. But I say that the isomorphisms are canonical. The MCG is not an abstract group, defined only up to automorphism, but a concretely represented group of isomorphisms. It is usually defined in terms of a surface with orientation. The orientation allows us to define the Dehn twist about a simple closed curve. The conjugacy class of the non-separating Dehn twist generates the torsion groups. Similarly, Dehn twists about boundaries allow us to write down canonical elements in the torsion-free case. $\endgroup$
    – Ben Wieland
    Oct 28, 2019 at 3:24
  • $\begingroup$ @BenWieland I agree that choosing an orientation on the surface (and a numbering of the punctures in case $(1,1)$) rigidifies things. What do you mean by 'the torsion groups" and "the torsion-free case"? $\endgroup$
    – YCor
    Oct 28, 2019 at 7:29

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