I want to know if I can have an inequivalent but weakly equivalent bundles over the Torus, i.e. if we can find explicitly an example of a bundle map $(\overline{f},f)$ where we require that $\overline{f}$ is an isomorphism from one fibre to another and $f$ an homeomorphism from the torus to the torus but such that $f$ is not the identity map (if this happens we will have an equivalence).
Notes
This restlessness is taken from Spivak's A comprehensive introduction to differential geometry,third edition,chapter 3.
I'm a beginner here so I don't know anything about cohomology, fancy differential geometry or that stuff thanks for your support.
Definition
A weak equivalence between two bundles over the same base space $B$ is a bundle map $(\overline{f},f)$ where $\overline{f}$ is an isomorphism on each fibre, and $f$ is a homeomorphism of $B$ into itself.