Foliations
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− | === | + | === Constructing new foliations from old ones === |
==== Pullbacks ==== | ==== Pullbacks ==== | ||
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{{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}} | {{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}} | {{cite|Conlon I}} | ||
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+ | ==== Glueing ==== | ||
+ | <wikitex>; | ||
+ | Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$. | ||
+ | If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$. | ||
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==== Turbulization ==== | ==== Turbulization ==== |
Revision as of 15:02, 7 June 2010
Contents |
1 Introduction
2 Construction and examples
2.1 Bundles
The most trivial examples of foliations are products , foliated by the leaves . (Another foliation of is given by .)
A more general class are flat -bundles. Given a representation , the flat -bundle with monodromy is given as , where acts on the universal cober by deck transformations and on by means of the representation . 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) mapThe fiber bundle yields a foliation by fibers . Its leaf space is (diffeomeorphic) homeomorphic to .
2.2 Submersions
This submersion yields a foliation of which is invariant under the -action given by for . The induced foliation of is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.3 Constructing new foliations from old ones
2.3.1 Pullbacks
[Conlon I]
2.3.2 Glueing
Let and be -manifolds with foliations of the same codimension. Assume there is a homeomorphism . If either both foliations are tangent or both foliations are transverse to the boundaries of and , then they can be glued to a foliation on . This called the tangential resp. the transversal glueing of and .
2.3.3 Turbulization
Let be a transversely orientable codimension 1 foliation, and let be an embedding transverse to .
Define a foliation on a small neighborhood by
The foliations and agree on a neighborhod of the boundary of . The result of glueing these foliations is called the turbulization of .
Invariants
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3 Classification/Characterization
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4 Further discussion
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5 References
This page has not been refereed. The information given here might be incomplete or provisional. |