Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix.
A fiber-preserving homeomorphism of $\xi$ is called automorphism of $\xi$. Each automorphism induces the mapping of the base space.
Can you help me to make a rigorous proof of following proposition? There exists an automorphism of $\xi$ that induces non-homotopic to identity base mapping if and only if the monodromy matrix of $\xi$ is conjugate to it's inverse.
The existence of such automorphism for suitable monodromy matrix is obvious. It's necessary that the monodromy matrix must be conjugate to it's inverse. To prove it, one can consider the fundamental group of the fiber. Unfortunately, my reasoning seem to be not enough rigorous.