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 $H_1(S_{0,n})$?


1 Answer 1


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.

  • $\begingroup$ I am exactly interested in the case when $g=0$ and $n \geq 3$. In this case $M_{0,n}$ is actually a manifold and the action of $\Gamma_{0,n}$ is free. In this specific case what would be the subgroup of $Out(\pi_1(S_{0,n})$ $\endgroup$ Oct 7, 2015 at 9:55
  • $\begingroup$ In the case of $n>0$, it's the subgroup that preserves the conjugacy classes that correspond to the boundary components. The book A primer on mapping class groups by Farb--Margalit is the definitive reference. $\endgroup$
    – HJRW
    Oct 7, 2015 at 13:14
  • $\begingroup$ I thought the mapping class in the genus zero case is the pure braid group in n-1 strands? $\endgroup$ Oct 7, 2015 at 13:31
  • $\begingroup$ $\Gamma_{0, \,n}$ is isomorphic to the mapping class group of the $n$-punctured sphere, see [J. S. Birman, Braid, Links and mapping class groups, Theorem 4.5] $\endgroup$ Oct 7, 2015 at 13:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.