Mapping Class Group of Pants with a Hole

Question

It is known that the mapping class group of the torus $\mathbb{T}^2$ is $\text{Mod}(\mathbb{T}^2) \cong \text{SL}_2(\mathbb{Z})$. We also know that for a pair of pants $P$ (a sphere with three boundary components), $\text{Mod}(P) \cong \mathbb{Z}^3$. Let $S$ be a pair of pants with a hole in it, i.e. a torus with three boundary components. Then what is $\text{Mod}(S)$? I think it should be something involving the above groups but I'm not sure how to go about finding it.

Answer 1

I'll assume you are defining mapping class groups of compact surfaces with nonempty boundary as in my comment.

The answer is that it's complicated, but, when $S$ is the torus with three boundary components then $\text{Mod}(S)$ can be described as a certain multiple extension group of $\text{SL}_2(\mathbb Z)$.

The first step is to write $\text{Mod}(S)$ as a certain central extension as follows. By collapsing each component of $\partial S$ to a point we obtain the torus $\mathbb T^2$ back again, together with a 3 point subset $P=\{p_1,p_2,p_3\} \subset \mathbb T^2$ in one-to-one correspondence with the components of $\partial S$. Let $\text{Mod}(\mathbb T^2;P)$ denote the mapping class group of homeomorphisms of $\mathbb T^2$ that fix each point of $P$, modulo isotopies that leave each point of $P$ stationary. Then we get a certain central extension $$1 \mapsto \mathbb Z^3 \mapsto \text{Mod}(S) \mapsto \text{Mod}(\mathbb T^2;P) \mapsto 1 $$ That's not a complete description, because to completely determine the central extension requires more information. But I'll shove that under the rug and continue.

For the second step, one uses the Birman short exact sequence to remove the special nature of the point $p_3$, obtaining a certain short exact sequence $$1 \mapsto \underbrace{\pi_1(\mathbb T^2 - \{p_1,p_2\})}_{\text{free of rank 3}} \mapsto \text{Mod}(\mathbb T^2;\underbrace{\{p_1,p_2,p_2\}}_{P}) \mapsto \text{Mod}(T^2;\{p_1,p_2\}) \to 1 $$ Again........ this is not a complete description without giving further information to determine the extension.........

For the third step, one again uses the Birman short exact sequence to remove the special nature of the point $p_2$, obtaining another short exact sequence $$1 \mapsto \underbrace{\pi_1(\mathbb T^2 - \{p_1\})}_{\text{free of rank 2}} \mapsto \text{Mod}(\mathbb T^2;\{p_1,p_2\}) \mapsto \text{Mod}(\mathbb T^2;\{p_1\}) \mapsto 1 $$ Again........ (and it's getting crowded under that rug) ........

For the final step, one uses the fact that $\mathbb T^2$ is a Lie group to obtain an isomorphism $$\text{Mod}(\mathbb T^2;\{p_1\}) \approx \text{Mod}(\mathbb T^2) \approx \text{SL}_2(\mathbb Z) $$