Question

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!

Answer 1

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 $\text{Mod}(T^2\setminus\{p\}):=\text{Homeo}^+(T^2\setminus \{p\})/\sim$.

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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 \[1\to \text{SL}_2(\mathbb{Z})\to \text{Homeo}(T^2\setminus \{p\})/\sim\to \mathbb{Z}/2\mathbb{Z}\to 1\]

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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: \[1\to \text{PMod}(T^2\setminus\{p,q\})\to \text{Mod}(T^2\setminus\{p,q\})\to \mathbb{Z}/2\mathbb{Z}\to 1\]

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A good reference for all these things is Farb-Margalit's "A Primer on Mapping Class Groups". In particular, the useful fact that there is no difference between homotopy and isotopy in dimension 2, or between considering homeomorphisms and diffeomorphisms, is covered in Chapter 1. The mapping class group of the torus is described in Chapter 2, starting with Theorem 2.15 on page 70.