$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p\_1,\dots, p\_n$: $\Mod(S^2\setminus \lbrace p\_1, \dots, p\_n\rbrace)$. I already know that the mapping class group of the disk $D^2$ minus $n$ points is the braid group $B\_n$, whose presentation is well known.


I already know a presentation of $\Mod(S^2\setminus \lbrace p\_1, \dots, p\_n\rbrace)$ and it can be found for example in the book


Benson Farb, Dan Margalit - A Primer on Mapping Class Groups (Princeton Mathematical)-Princeton University Press (2011), page 258, Chapter 9.


However, I don't understand from there how to get the usual presentation if this group, which is


$$\langle \sigma\_1, \dots, \sigma\_{n-1} | rel. of B\_n, (\sigma\_1,\dots, \sigma\_{n-1})^n, \sigma\_1\dots\sigma\_{n-2}\sigma\_{n-1}^2\sigma\_{n-2}\dots \sigma\_1 \rangle. $$


Thus the question is:


How can use the mapping class group $\Mod(D^2)\cong B\_n$ to get the above presentation of $S^2\setminus \{p\_1,\dots, p\_n\}$?


An idea could be also sufficient :).



How to get a presentation of the mapping class group of the $n$-punctured sphere
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Emb{Emb}\DeclareMathOperator\fix{fix}$There's a variety of ways to do this. If you take "mapping class group" to mean "isotopy classes of diffeomorphisms" then a fairly natural approach is to consider the braid group on $n$ strands to be $\pi\_0$ of diffeomorphisms of $S^2$ that preserve $n$ points and fix a disc pointwise. The mapping class group of the sphere with $n$ marked points removes this disc constraint. You have a fibre sequence


$$ \Diff(S^2, n, \fix D^2) \to \Diff(S^2, n) \to \Emb(D^2, S^2 \setminus n) $$


You get the presentation you seek by looking at the homotopy long exact sequence, and assembling the associated short exact sequence for $\pi\_0 \Diff(S^2, n)$. This notation again means isotopy classes of diffeomorphisms of $S^2$ that preserves $n$ points, as a set.


The space of embeddings of $D^2$ in the $n$-times punctured $S^2$ has the homotopy-type of $SO\_2 \times (S^2 \setminus n)$, i.e. a circle cross a wedge of circles.


