A fibration over the circle as a globally stable foliation

The integer-linear map $(x,y)\mapsto (x+y,y)$ descends to the torus, and $(x,y)\mapsto (x-y,y)$ is its smooth two-sided inverse. Thus F1 applies and supplies the smooth mapping torus, fibre bundle, torus leaves and trivial holonomy. The form $dt$ is invariant under the quotient action and coorients the fibre foliation.

The map $f$ fixes the horizontal coordinate loop and sends the vertical loop to the loop $(s,s)$, the sum of the two generators in F2. Its induced matrix is $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, which is not the identity. An isotopy to the identity would induce the identity on the abelian fundamental group: the basepoint motion changes induced maps only by conjugation, and conjugation in ${\mathbb{Z}}^2$ is trivial. Therefore $f$ is not isotopic to the identity, and F1 shows that the bundle is not isomorphic over $S^1$ to the product bundle. F2 also shows that every leaf has infinite fundamental group, so no finite-fundamental-group instance of the global theorem has been asserted.

Starting at $[z,0]$ and flowing positively in $t$ for time one gives $[z,1]=[f^{-1}(z),0]$. Thus the whole-fibre return is $f^{-1}$, with no sign-convention ambiguity.

This is a global compact holonomy-free torus foliation with nontrivial bundle monodromy and positive return $f^{-1}$; all conclusions were verified from the explicit quotient and mapping-torus supplier under countable choice.