# Are there results about the group of homeomorphisms of $(T^2-\{*,*\})$ up to isotopy?

I am studying a fiber bundle over circle with fiber $T^2-\{*,*\}$. Since this is a mapping torus, the group $Homeo(T^2-\{*,*\})/isotopy$ plays an important role. Are there some existing theorems on this field? Thank you!

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Your original title was obscure. I've edited it to something more explicit. I hope the new title still represents your question – Charles Rezk Mar 26 '10 at 16:33

There are many results in this field, and such groups, called mapping class groups, are well-studied. In the case of a torus the situation is totally understood; the mapping class group of the torus is $\text{SL}_2(\mathbb{Z})$. The only problem is that I am not sure what $\{*,*\}$ means. Note: the group $\text{Homeo}(T^2\setminus \{p\})/\sim$ is called the extended mapping class group, denoted $\text{Mod}^{\pm}(T^2\setminus\{p\})$, while the subgroup of orientation-preserving homeomorphisms is the mapping class group $\text{Mod}(T^2\setminus\{p\}):=\text{Homeo}^+(T^2\setminus \{p\})/\sim$.
If you mean a one-element set, something like $\{(0,0)\}$: in the case of a torus, the missing point turns out not to matter: $\text{Mod}(T^2\setminus\{p\})=\text{Mod}(T^2)$. This group, of orientation-preserving homeomorphisms up to isotopy, is isomorphic to $\text{SL}_2(\mathbb{Z})$. Your group is then an extension of $\mathbb{Z}/2\mathbb{Z}$ by this group, corresponding to the action on the orientation. $$1\to \text{Mod}(T^2\setminus\{p\})\to \text{Mod}^{\pm}(T^2\setminus\{p\})\to \mathbb{Z}/2\mathbb{Z}\to 1$$ which can be written as $$1\to \text{SL}_2(\mathbb{Z})\to \text{Homeo}(T^2\setminus \{p\})/\sim\to \mathbb{Z}/2\mathbb{Z}\to 1$$
If you mean a two-element set, then first consider the subgroup $\text{PMod}(T^2\setminus\{p,q\})$ of homeomorphisms that don't "switch" the two punctures. The map given by "filling in the puncture $q$" gives an extension $$1\to\pi_1(T^2\setminus\{p\},q) \to\text{PMod}(T^2\setminus\{p,q\})\to \text{Mod}(T^2\setminus\{p\})\to 1$$ which can also be written $$1\to F_2\to\text{PMod}(T^2\setminus\{p,q\})\to \text{SL}_2(\mathbb{Z})\to 1$$ since $\pi_1(T^2\setminus\{p\},q)$ is a free group of rank two. The mapping class group is an extension of $\mathbb{Z}/2\mathbb{Z}$ by this group, corresponding to whether the punctures are switched: $$1\to \text{PMod}(T^2\setminus\{p,q\})\to \text{Mod}(T^2\setminus\{p,q\})\to \mathbb{Z}/2\mathbb{Z}\to 1$$