F-test of equality of variances: Difference between revisions

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In [[mathematics]], '''Novikov's compact leaf theorem''', named after [[Sergei Novikov (mathematician)|Sergei Novikov]], states that
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: ''A codimension-one [[foliation]] of a compact 3-manifold whose [[universal covering space]] is not contractible must have a compact leaf.''
 
== Novikov's compact leaf theorem for ''S''<sup>3</sup> ==
Theorem: ''A smooth codimension-one foliation of the [[3-sphere]]'' ''S''<sup>3</sup> ''has a compact leaf. The leaf is a torus'' ''T''<sup>2</sup> ''bounding a [[solid torus]] with the [[Reeb foliation]].''
 
The theorem was proved by [[Sergei Novikov (mathematician)|Sergey Novikov]] in 1964. Earlier [[Charles Ehresmann]] had conjectured that every smooth codimension-one foliation on ''S''<sup>3</sup> had a compact leaf, which was true for all known examples; in particular, [[Reeb foliation]] had a compact leaf that was&nbsp;''T''<sup>2</sup>.
 
== Novikov's compact leaf theorem for any ''M''<sup>3</sup> ==
In 1965, Novikov proved the compact leaf theorem for any&nbsp;''M''<sup>3</sup>:
 
Theorem: ''Let'' ''M''<sup>3</sup> ''be a closed 3-manifold with a smooth codimension-one foliation''&nbsp;''F''. ''Suppose any of the following conditions is satisfied:''
 
# the ''[[fundamental group]] <math>\pi_1(M^3)</math> is finite,''
# the ''second [[homotopy group]] <math>\pi_2(M^3)\ne 0</math>,''
# ''there exists a leaf <math>L\in F</math> such that the map <math>\pi_1(L)\to\pi_1(M^3)</math> induced by inclusion has a non-trivial [[Kernel (algebra)|kernel]].
''
 
''Then'' ''F'' ''has a compact leaf of [[genus]]'' ''g''&nbsp;≤&nbsp;1.
<!--
In fact, except in case (2), where the closed leaf might be ''S''<sup>2</sup> or <math>RP^2</math>, in all other cases, the foliation contains a Reeb component.
-->
 
In terms of covering spaces:
 
''A codimension-one [[foliation]] of a compact 3-manifold whose [[universal covering space]] is not contractible must have a compact leaf.  
''
 
==References==
* ''S. Novikov''. The topology of foliations//Trudy Moskov. Mat. Obshch, 1965, v. 14, p. 248&ndash;278.[http://www.mi.ras.ru/~snovikov/23.pdf]
* ''I. Tamura''. Topology of foliations — AMS, v.97, 2006.
* ''D. Sullivan'', Cycles for the dynamical study of foliated manifolds and complex manifolds, ''Invent. Math.'', '''36''' (1976), p. 225&ndash;255. [http://www.kryakin.com/files/Invent_mat_%282_8%29/36/36_08.pdf]
 
[[Category:Foliations]]
[[Category:Theorems in topology]]
{{topology-stub}}

Revision as of 11:12, 23 February 2014

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