Let $T\_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M\_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the mapping class group $\Gamma\_{0,n}$, so that $M\_{0,n} \cong T\_{0,n} / \Gamma\_{0,n}$? 


Now $\pi\_1(M\_{0,n}) \cong \Gamma\_{0,n}$ since the Teichmuller space is contractible and the action of $\Gamma\_{0,n}$ is free.


What does the Dehn-Nielsen-Baer theorem say in this case? Let 
$S\_{0,n}$ be the $n$ punctured complex projective line then which subgroup of $Out(\pi\_1(S\_{0,n}))$ does $\Gamma\_{0,n}$ correspond to? Also how does $\Gamma\_{0,n}$ act on $H\_1(S\_{0,n})$?



Mapping class group of a punctured genus 0 surface
Let $S\_g$ be a compact Riemann surface of genus $g$ with $n$ marked points $x\_1, \ldots, x\_n$, and set $S\_{g, \, n}:=S\_g - \{x\_0, \ldots, x\_n\}$. Also, denote by $\pi\_{g, \, n}$ the fundamental group of $S\_{g, \, n}$ (we omit the decoration $n$ when $n=0$).


Then the mapping class group $\Gamma\_{g,\, n}$ is defined as $$\Gamma\_{g, \, n} := \pi\_0 \, \textrm{Diff}^{+}(S\_{g, \, n}),$$
where $\textrm{Diff}^{+}(S\_{g, \, n})$ is the group of orientation-preserving diffeomorphisms of $S\_g$ that fix each $x\_i$.


The group $\Gamma\_{g, \, n}$ naturally acts on the Teichmueller space $T\_{g, \, n}$. Such an action is properly discontinuous and there is a subgroup of finite index acting freely, in such a way that, as an orbifold, $$M\_{g, \, n}= T\_{g, \, n}/ \Gamma\_{g, \, n}.$$
In particular, $\pi^{\textrm orb}\_1(M\_{g, \, n}) = \Gamma\_{g, \, n}$.


By a result of Baer and Nielsen, with this representation $\Gamma\_{g}$ is identified with the subgroup of $\textrm{Out}(\pi\_{g})$ acting trivially on $H\_2(\pi\_{g})$. When $n \geq 1$, there are similar characterizations. 


For (many) more details, see the paper


R. Hain, E. Looijenga: *Mapping class groups and moduli spaces of curves,* Algebraic Geometry-Santa Cruz 1995, 97–142, Proc. Sympos. Pure Math., **62**, Part 2, Amer. Math. Soc., Providence, RI (1997),


also available [on the arXiv](http://arxiv.org/abs/alg-geom/9607004).


