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$\DeclareMathOperator\Mod{Mod}$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 :).

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    $\begingroup$ Your reverted an edit which made "rel. of" text mode. Instead, it is ill-formated as if it were variables $r$,$e$, $l$, $o$, $f$. Also \mid makes correct spacing when $|$ makes wrong spacing. $\endgroup$
    – YCor
    Mar 13 at 18:45

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$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Emb{Emb}\DeclareMathOperator\fix{fix}$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.

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  • $\begingroup$ What do you mean diffeomorphisms that preserves $n$ points and fix a disc point wise? Which disk do you mean? Can you go more in details please? $\endgroup$
    – Federico Fallucca
    Mar 13 at 19:34
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    $\begingroup$ The disc is any embedded disc in $S^2$ disjoint from the $n$ points, i.e. it does not matter because any two embedded copies of a disc with $n$ disjoint points in $S^2$ are isotopic. I believe this argument appears in Birman's book, perhaps not in exactly this form but this general idea likely appears there. $\endgroup$
    – Ryan Budney
    Mar 13 at 20:21

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