Topology and Foliations

  • #1
I was suprised to realize that foliation theory was actually closely related to topology. Indeed, http://www.map.mpim-bonn.mpg.de/Foliations, states a theorem which say that a codimension one foliation exists if and only if the Euler characteristic of the topological space is one!

I am currently working with foliations, so I wondered if someone here knew the consequences of both a codimesion one foliation _and_ a one-dimensional foliation existing on the same time?

What does it imply for the topology?

The first type of foliation is often known as hypersurface foliations (or slicing), while the second is known also known as threading.
 

Answers and Replies

  • #2
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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
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your question is a little vague, but here goes. what are the open sets of a smooth curve in R^3 (for example)? then what is a differentiable map from this curve into R^1? what does differentiability imply about continuity?
 
  • #4
lavinia
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Take any compact manifold without boundary and take its Cartesian product with a circle. The product has a codimension 1 foliation but it's Euler characteristic is zero.
 
Last edited:
  • #5
WWGD
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From what I remember, if you have a k-plane bundle of fiber F ## M \rightarrow B ## , its Euler class is ## \chi (F) PD(M) ## , where ## \chi ## is the Euler characteristic and PD is the Poincare dual.
 

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