## orientation preserving diﬀeomorphism on T^n [closed]

Given $S \in GL_n(\mathbb{R}^n)$.

Show that $x \mapsto Sx$ is an orientation preserving diﬀeomorphism on $\mathbb{T}^n$ if and only if $S \in SL_n(\mathbb{Z}^n)$.

I'm working on the only if part. Orientation perserving implies $\det(S) > 0$.

How can I prove that we must have $S \in SL_n(\mathbb{Z}^n)$. I can see that when $S \in SL_n(\mathbb{R}^n)$ is not enough to be surjective, but can´t prove that $S \in SL_n(\mathbb{Z}^n)$.

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I imagine that your assignment is somehow induced by a map on the first homology of your n-torus. However, the group $SL_2(\mathbb{Z}$ is countable while the diffeomorphisms of the circle are uncountable. I get the feeling that you are interested in the mapping class group of the n-torus. – Spice the Bird Dec 4 at 16:39
I´m sorry I not really follow you. I was hoping for a simple linear/algebra prove. – unknown (google) Dec 4 at 17:34
You say you're working on the if part, but what you wrote after that is entirely about the "only if" part, i.e., the left-to-right implication. – Andreas Blass Dec 4 at 18:27
You right Andreas, my mistake, I changed it. – unknown (google) Dec 4 at 18:54
You might be better off asking this question on math.stackexchange.com where it will probably receive a decent answer in a reasonable amount of time. – Martin Dec 4 at 21:36