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I'm reading Automorphisms of Surfaces after Nielsen and Thurston by Casson and Bleiler, and I'm confused by something in the introductory section where the authors classify the homeomorphisms of a torus.

They say that, "The homeomorphisms of $T^2$ correspond to the elements of the general linear group $GL_2(\mathbb{Z})$ as any element $\alpha$ in $GL_2(\mathbb{Z})$ maps $\mathbb{Z}^2$ to itself and so induces a continuous map $h_\alpha:T^2\to T^2$."

I understand why the map $h_\alpha$ is induced by $\alpha$, but it's not clear to me why every homeomorphism of the torus is induced by some element of $GL_2(\mathbb{Z})$. Any homeomorphism $f:T^2\to T^2$ certainly lifts to an invertible map $\tilde f:\mathbb{R}^2\to\mathbb{R}^2$, but why is $\tilde f$ linear?

Nathan Lopez
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    Looks like a misunderstanding. Clearly, there are many more homeomorphisms, uncountably many. Are you sure they didn't factor out some relation, such as "up to isotopy" or so? – Peter Franek Dec 18 '16 at 22:32
  • Definitely possible. But I don't see that anywhere in the introduction, which is only 2 pages long. The quote I have above is a direct quote, no paraphrasing, and I figure their use of the word 'correspond' wasn't supposed to be colloquial. I can link to a picture of their work if that would be helpful (and if stack exchange rules allow it). – Nathan Lopez Dec 18 '16 at 22:42
  • Maybe it was a "high-level language" and "corresponds" didnd't really mean a 1-1 correspondence... – Peter Franek Dec 18 '16 at 22:43
  • Yea, maybe. Here's a link to the introduction if you're willing and interested to read it for yourself – Nathan Lopez Dec 18 '16 at 22:55
  • http://imgur.com/a/JqhI8 – Nathan Lopez Dec 18 '16 at 22:55
  • They are referring to Nielsen–Thurston classification so you should view it in this context. – user60589 Dec 19 '16 at 09:47
  • I'm not sure I understand what you're saying 60589. The book is about the NT classification, yes, but how does that inform this part? Could you be more explicit please? – Nathan Lopez Dec 19 '16 at 19:25
  • I happened to see this post and would like to add a comment, just for the ones who see this later... Since $\mathbb{T}^2$ is a $K(G,1)$ space, the homotopy classes of self-homeomorphisms are in one-to-one correspondence with $\text{Aut}(\pi_1\mathbb{T}^2)\simeq GL(2,\mathbb{Z})$, and homotopy on a surface implies isotopy, which is in comparison as the Nielsen-Thurston classification the user mentioned above. (I'm just guessing what he or she means...) – ah-- Sep 22 '19 at 20:51

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