Mapping torus foliations realize global Reeb stable examples

(The action is free, properly discontinuous and foliation-preserving.) For $m,n\in \mathbb{Z}$ one has $m\cdot (n\cdot (x,t))=(f^m(f^n(x)),t+n+m)=(f^{m+n}(x),t+m+n)$, so the formula defines an action [F3, F4]. It is free: an element $n$ with $n\cdot (x,t)=(x,t)$ forces $t+n=t$, hence $n=0$. It is properly discontinuous: for a compact $K\subseteq L\times \mathbb{R}$ the set of $n$ with $(K+(0,n))\cap K\mathrm{\ne }\mathrm{\varnothing }$ is finite, since the $t$-coordinates must satisfy $\mathrm{\mid }n\mathrm{\mid }\le$ the diameter bound of $K$ in the $t$-direction. Finally it preserves the product foliation: $n\cdot (L\times \{t\})=L\times \{t+n\}$ and the restriction is the diffeomorphism $f^n$ [F3, F5].

(Quotient manifold and foliation.) By [F1] the quotient $M=(L\times \mathbb{R})\mathrm{/}\mathbb{Z}$ carries a unique smooth structure making the orbit map a local diffeomorphism (and hence a covering map), and the product foliation descends to the regular codimension-one foliation $F_f$ whose leaves are the images of the slices $L\times \{t\}$ [F1]. The closed manifold $L$ is compact without boundary, and $L\times [0,1]$ is a compact fundamental domain for the action, so $M$ is compact without boundary: $M$ is a closed smooth manifold.

(Leaves and holonomy.) Each slice $L\times \{t\}$ is compact and the action carries slices diffeomorphically onto slices, so every leaf of $F_f$ is a compact manifold diffeomorphic to $L$ [F1, F5]. The holonomy of a leaf is trivial: no nonzero deck transformation stabilizes a slice, since it changes $t$ by a nonzero integer. Thus a leafwise loop lifts to a closed loop in that slice, whose transverse transport is the identity in the product structure [F6].

(Bundle structure.) The projection $q_0(x,t):=t<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>1$ satisfies $q_0(n\cdot (x,t))=t+n<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>1=q_0(x,t)$, hence descends to a smooth map $q:M\to S^1$ whose fibres are exactly the images of the slices, that is, the leaves [F1, F5]. Over an interval $I\subseteq S^1$ the identification $q^{-1}(I)\cong L\times I$ is a diffeomorphism commuting with $q$, so $q$ is a locally trivial fibre bundle with fibre $L$ whose fibre foliation is $F_f$ [F5].

(Transverse direction and monodromy.) The vector field $\mathrm{\partial }\mathrm{/}\mathrm{\partial }t$ on $L\times \mathbb{R}$ is invariant under the action and transverse to the slices, so it descends to a nowhere-vanishing vector field transverse to $F_f$ [F1, F5]. Its flow after one unit of time sends $(x,0)$ to $(x,1)$, and $(x,1)$ is equivalent under the action to $-1\cdot (x,1)=(f^{-1}(x),0)$; therefore the first-return map of the descended flow on the fibre over $q(L\times \{0\})$ is $x\mapsto f^{-1}(x)$. This is the monodromy for the displayed quotient convention.

(Bundle structures over the fixed oriented circle.) Cut the base at a point. A finite interval subdivision subordinate to product charts trivializes the pullback bundle over $[0,1]$: successively modify each next trivialization by its overlap transition, extending that transition along the interval by a smooth reparametrization constant near the joining endpoint. The remaining endpoint identification is a fibre diffeomorphism $g$. If an isomorphism over the fixed oriented base identifies two such gluings $g,g^{\mathrm{\prime }}$, its interval trivializations give a path $h_t$ of fibre diffeomorphisms satisfying $h_{0}g=g^{\mathrm{\prime }}{h}_1$. Thus the mapping classes of $g,g^{\mathrm{\prime }}$ are conjugate. Conversely, if their mapping classes are conjugate, choose $h_0$ giving that conjugation and an isotopy from $h_0$ to $h_1=(g^{\mathrm{\prime }}{)}^{-1}{h}_{0}g$, constant near endpoints. The map $(y,t)\mapsto (h_t(y),t)$ respects the endpoint identifications and descends to a bundle isomorphism. Hence bundles over this fixed base are classified by conjugacy classes in ${\pi }_0\mathrm{Diff}(L)$. A conjugacy class equals the identity class precisely when $g$ is isotopic to the identity. Here $g=f^{-1}$ by step 1.5, so the bundle is trivial exactly when $f$ is isotopic to the identity; distinct nonconjugate mapping classes give distinct bundle structures. No assertion about the bare total manifold is made.