  In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and
$\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null
homotopic loop $\{\Phi_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that
$\Phi_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $\Phi_0=\Phi_1= Id$.
As a direct consequence, we get a general process for building new partially
hyperbolic diffeomorphisms on closed $3$-manifolds. \cite{BPP} built a new
example of dynamically coherent non-transitive partially hyperbolic
diffeomorphism on a closed $3$-manifold, the example in \cite{BPP} is obtained
by composing the time $t$ map, $t>0$ large enough, of a very specific
non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result
shows that the same construction holds starting with any non-transitive Anosov
flow on an oriented $3$-manifold. Moreover, for a given transverse torus, our
result explains which type of Dehn twists lead to partially hyperbolic
diffeomorphisms.
