Lift of a diffeomorphism of the Torus I'm trying to prove the following formula. Suppose to have $p:\mathbb{R}^{d}\rightarrow\mathbb{T}^{d}$
  the canonical projection of the real d- dimensional space in to the d-dimensional torus, and suppose to have a diffeomorphism (or more in genearl an homeomorphism) $\varphi:\mathbb{T}^{d}\longrightarrow\mathbb{T}^{d}$
  that i can lift to a diffeomorphism $\tilde{\varphi}:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$
  such that $p\circ\tilde{\varphi}=\varphi\circ p$
 . It's true that i can write like $\tilde{\varphi}\left(x\right)=Ax+u\left(x\right)$
  where $A\in Sl(\mathbb{Z},d)$
  and $u\left(x\right)$
 is periodic ?
I have proved that if A
  is a lift of a linear diffeomorphism of the torus than it lies in $Sl(\mathbb{Z},d)$.
  Any suggestion for the rest?
 A: If you have a continuous $\varphi \colon \mathbb{T}^d \to \mathbb{T}^d$ and a lift $\tilde{\varphi} \colon \mathbb{R}^d \to \mathbb{R}^d$ of $\varphi$, then for every $z \in \mathbb{Z}^d$, the function $\psi_z\colon \mathbb{R}^d \to \mathbb{R}^d; \: \psi_z(x) = \tilde{\varphi}(x+z) - \tilde{\varphi}(x)$ is continuous and $\mathbb{Z}^d$- valued, since $$p(\tilde{\varphi}(x+z)) = \varphi(p(x+z)) = \varphi(p(x)) = p(\tilde{\varphi}(x)),$$ hence constant. It is easy to see that $z \mapsto \psi_z(0) = \tilde{\varphi}(z) - \tilde{\varphi}(0)$ is $\mathbb{Z}$-linear, so we can write $\psi_z(0) = Az$ with a matrix $A \in M_d(\mathbb{Z})$. Then $u \colon \mathbb{R}^d \to \mathbb{R}^d;\; u(x) = \tilde{\varphi}(x) - Ax$ is $\mathbb{Z}^d$-periodic, since
$$u(x+z) = \tilde{\varphi}(x+z) - A(x+z) = \tilde{\varphi}(x) + \psi_z(x) - Ax - Az = u(x) + \bigl(\psi_z(x) - Az\bigr) = u(x).$$
If $\varphi$ is a homeomorphism, then $\tilde{\varphi}\bigl([0,1)^d\bigr)$ is a fundamental region of the lattice $\mathbb{Z}^d$, hence then $A \in GL_d(\mathbb{Z})$.
