Foliations

1 Introduction

Let $M$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ foliation of $M$ if $M$ admits an atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is charts (diffeomorphisms) $$\phi_\alpha:U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. </wikitex> == Construction and examples == === Bundles === <wikitex>; The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.) A more general class are flat $Diff\left(F\right)$-bundles. Given a representation $\pi_1B\rightarrow Diff\left(F\right)$, the flat $Diff\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers. Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative: $$\begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\ U\ar[r]^{id}&U} \end{xy}$$ The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its leaf space is (diffeomeorphic) homeomorphic to $B$. </wikitex> === Submersions === <wikitex>; Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. An example of a submersion, which is not a fiber bundle, is given by $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-action given by $z\left(x,y\right)=\left(x,y+z\right)$ for $\left(x,y\right)\in\left[[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)=\left[-1,1\right]\times S^1$ is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff. </wikitex> === Constructions === ==== Pullbacks ==== <wikitex>; {{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}} {{cite|Conlon I}} ==== Turbulization ==== <wikitex>; Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by $$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$ where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$ The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhod of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$. == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}M$ be an $n$$n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$$M$ into connected, topologically immersed submanifolds of dimension $n-q$$n-q$. ${\mathcal{F}}$${\mathcal{F}}$ is said to be a codimension $q$$q$ foliation of $M$$M$ if $M$$M$ admits an atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$$\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is charts (diffeomorphisms)

$\displaystyle \phi_\alpha:U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$ $$\displaystyle \phi_\alpha:U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$

such that for each $\alpha\in{\mathcal{A}}, b\in B$$\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U\alpha$$F_b\cap U\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$$\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.

2 Construction and examples

2.1 Bundles

The most trivial examples of foliations are products $M=B\times F$$M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$$F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$$M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$$B_f:=\left\{f\right\}\times B, f\in F$.)

A more general class are flat $Diff\left(F\right)$$Diff\left(F\right)$-bundles. Given a representation $\pi_1B\rightarrow Diff\left(F\right)$$\pi_1B\rightarrow Diff\left(F\right)$, the flat $Diff\left(F\right)$$Diff\left(F\right)$-bundle with monodromy $\rho$$\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$$M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$$\pi_1B$ acts on the universal cober $\widetilde{B}$$\widetilde{B}$ by deck transformations and on $F$$F$ by means of the representation $\rho$$\rho$. A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers.

Flat bundles fit into the frame work of fiber bundles. A (smooth) map

$\displaystyle \pi:M\rightarrow B$ $$\displaystyle \pi:M\rightarrow B$$

between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$$b\in B$ has an open neighborhood $U$$U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$$\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$$p_1$ projection to the first factor) commutative:

$\displaystyle \begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ U\ar[r]^{id}&U} \end{xy}$

The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$$F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its leaf space is (diffeomeorphic) homeomorphic to $B$$B$.

2.2 Submersions

Let

$\displaystyle f:M\rightarrow B$ $$\displaystyle f:M\rightarrow B$$

be a submersion. Then $M$$M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$$\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. An example of a submersion, which is not a fiber bundle, is given by

$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$ $$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$

$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$ $$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$$

This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$$\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$${\mathbb Z}$-action given by $z\left(x,y\right)=\left(x,y+z\right)$$z\left(x,y\right)=\left(x,y+z\right)$ for $\left(x,y\right)\in\left[[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$$\left(x,y\right)\in\left[[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)=\left[-1,1\right]\times S^1$${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)=\left[-1,1\right]\times S^1$ is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff.

2.3 Constructions

2.3.1 Pullbacks

Theorem 2.1. If $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$$q$ and $f:N\rightarrow M$$f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$$\mathcal{F}$, then $N$$N$ is foliated by connected components of $f^{-1}\left(L\right)$$f^{-1}\left(L\right)$ as $L$$L$ ranges over the leaves of $\mathcal{F}$$\mathcal{F}$.

[Conlon I]

Turbulization

Let $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$$\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$$\mathcal{F}$.

Define a foliation ${\mathcal{F}}_0$${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$$N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by

$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$ $$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$

where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$$\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with

$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$ $$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$

The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$$\mathcal{F}_0$ agree on a neighborhod of the boundary of $N\left(\gamma\left(S^1\right)\right)$$N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$${\mathcal{F}}^\prime$ of $\mathcal{F}$$\mathcal{F}$.

Invariants

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3 Classification/Characterization

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4 Further discussion

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5 References

• [Conlon I] Template:Conlon I

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