Reeb Stability and Global Foliation Constructions — Examples

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.

(The quotient is the Möbius band.) The map $\tau$ is an involution: ${\tau }^2(u,v)=\tau (u+\frac{1}{2},-v)=(u+1,v)=(u,v)$ in $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$. It is free: $\tau (u,v)=(u,v)$ would give $v=-v$, hence $v=0$, and then $u=u+\frac{1}{2}$, impossible in $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$; since $S^1\times (-1,1)$ is a surface and the action is free and properly discontinuous, the quotient $M$ is a smooth surface. It is the total space of the interval bundle over $S^1=\mathbb{R}\mathrm{/}(\frac{1}{2}\mathbb{Z})$ with monodromy $v\mapsto -v$, the non-trivial interval bundle, i.e. the Möbius band.

(The foliation descends and the holonomy is ${\mathbb{Z}}_2$.) The involution carries the slice $S^1\times \{v\}$ onto $S^1\times \{-v\}$, so the product foliation is preserved and descends to a codimension-one foliation of $M$ whose leaves are the images of the slices [F1]. The image of $S^1\times \{0\}$ is the central leaf $L_0=\mathbb{R}\mathrm{/}(\frac{1}{2}\mathbb{Z})$; following it once around means passing from $u$ to $u+\frac{1}{2}$, and the identification in the quotient returns the transversal coordinate $v$ to $-v$, so the return germ is the reflection $v\mapsto -v$ and the holonomy group is ${\mathbb{Z}}_2$ [F2].

(The holonomy cover and the normal model.) The holonomy cover of $L_0$ is the connected double cover ${\widehat{L}}_0=S^1\to L_0$ with deck group ${\mathbb{Z}}_2$ acting by $u\mapsto u+\frac{1}{2}$ [F2]. With $D$ a small invariant transversal interval and the diagonal action $(u,v)\mapsto (u+\frac{1}{2},-v)$, the finite-holonomy normal model $({\widehat{L}}_0\times D)\mathrm{/}{\mathbb{Z}}_2$ is the quotient of $S^1\times D$ by exactly the involution $\tau$ restricted to $S^1\times D$, hence equals the Möbius band over $D$ with the descended foliation; the central leaf is $({\widehat{L}}_0\times \{0\})\mathrm{/}{\mathbb{Z}}_2\cong L_0$ [F3].

(The other leaves.) For $v\mathrm{\ne }0$, the map $u\mapsto [(u,v)]$ from $S^1$ to the quotient leaf is injective: the only nonidentity group element changes $v$ to $-v$. It parametrizes that entire leaf, because the other slice at $-v$ has the same image. Under the model retraction to $L_0=\mathbb{R}\mathrm{/}(\frac{1}{2}\mathbb{Z})$ its projection is $u<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathbb{Z}\mapsto u<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>(\frac{1}{2}\mathbb{Z})$, a double covering. In a fixed local transverse fibre the two leaf intersections at $v$ and $-v$ are distinct points; they are not identified in that fibre. This is exactly the stabilizer calculation $[H:H_v]=2$ for $H_v=\{1\}$.

(A compact leaf.) By [F1] the boundary $\mathrm{\partial }X$ is a single leaf of the Reeb foliation and it is compact, diffeomorphic to the two-torus $T^2$.

(Boundaryless extension and infinite holonomy.) Near $r=1$, the interior level foliation of F1 has kernel $dr-v(r)dt$, where $v(r)=(1-r^2{)}^2\exp (-1\mathrm{/}(1-r^2))\mathrm{/}(2r)$. Extend $v$ by zero for $r\ge 1$. It is smooth and flat at one: every derivative is a smooth factor times a polynomial in $(1-r^2{)}^{-1}$ times the rapidly decaying exponential. On the added collar in $Y=\{r<1+\epsilon \}\times S^1$, the form $\alpha =dr-v(r)dt$ is nowhere zero and satisfies $\alpha \wedge d\alpha =0$. Its kernel agrees with the original foliation inside the collar, so F5 supplies a boundaryless extension with $r=1$ still a torus leaf. A radial interval at fixed angles is now a two-sided transversal. The holonomy of the loop in the $S^1$-factor of the Reeb component is computed in [F1] as the germ $r\mapsto r^{\mathrm{\prime }}$ determined by $u(r^{\mathrm{\prime }})=u(r)+1$; it is a non-identity one-sided contraction, so the holonomy group contains a non-identity element, and its iterates $u(r_n)=u(r)+n$ give infinitely many distinct germs near the boundary. These distinct inward restrictions force the two-sided germs in the extension to be distinct, so its compact torus leaf also has infinite holonomy in the exact sense of F2.

(The leaf is not stable.) Suppose the boundary leaf were stable. Then a small collar $W=\{r>1\mathrm{/}2\}$ of it would contain a saturated neighbourhood $U$ of the boundary leaf [F3]; but $U$ is open and contains the boundary leaf, hence contains a point $p$ with $r(p)$ close to $1$, and being saturated it contains the entire leaf through $p$; by [F1] that leaf is a plane, and it meets the circle $r=1\mathrm{/}2$ and so is not contained in $W$, a contradiction. Therefore the compact boundary leaf is not stable.

(What this refutes.) A compact leaf can have infinite holonomy and can fail to be stable, so the finiteness hypothesis of [F4] cannot be dropped for compact leaves, and the failure is a failure of finiteness of holonomy, not of compactness of the ambient foliated manifold.

(Leaves and their holonomy.) The slices $S^1\times \{t\}$ are the maximal connected integral manifolds of the kernel of $dt$, hence the leaves of the product foliation [F1]. A leafwise loop lies inside a single slice $L_t$, and following it transports the vertical transversal $\{u\}\times (t-\epsilon ,t+\epsilon )$ by the identity in product coordinates, since the second coordinate is constant along the slices; hence the holonomy representation of every leaf is trivial [F2].

(Product neighbourhoods.) Fix $t_0$ and $\delta >0$. The set $U_{\delta }=S^1\times (t_0-\delta ,t_0+\delta )$ is open, contains $L_{t_0}$, and is a union of slices, hence saturated; the translation $(u,t)\mapsto (u,t-t_0)$ is a foliated diffeomorphism onto $S^1\times (-\delta ,\delta )$ with the product foliation, and these neighbourhoods for shrinking $\delta$ form a fundamental system. This is exactly the conclusion of the product corollary for a compact leaf of trivial holonomy [F3].

(The torus variant.) Replacing $S^1$ by $T^2$ in the same argument gives the product foliation of $T^2\times \mathbb{R}$: the slices are compact leaves with trivial holonomy by the same computation, while their fundamental group is ${\mathbb{Z}}^2$, which is infinite [F4]. Hence trivial holonomy does not require finiteness of ${\pi }_1$, and the same direct product calculation applies to every nonempty connected closed smooth fibre $L$, independently of its fundamental group.

(The leaves and their fundamental groups.) The slices $T^2\times \{\theta \}$ are the leaves of the product foliation, each compact and diffeomorphic to $T^2$ [F1]; the fundamental group of every leaf is ${\pi }_1(T^2)\cong {\mathbb{Z}}^2$, which is infinite [F2].

(Trivial holonomy.) A local transversal is a vertical circle segment $\{p\}\times (-\epsilon ,\epsilon )$, and the leafwise transport in product coordinates is the identity because the second coordinate is constant on the slices; hence the holonomy representation of every leaf is trivial [F3].

(Stability and what this refutes.) Since the leaves are compact with trivial holonomy, [F4] gives a fundamental system of saturated product foliated neighbourhoods $T^2\times (-\epsilon ,\epsilon )$ of every leaf, so every leaf is stable, while its fundamental group is infinite. Hence a compact leaf can have trivial (in particular finite) holonomy and be stable although its fundamental group is infinite, so finiteness of the fundamental group is not equivalent to finiteness of holonomy and is not necessary for stability.