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Are there examples of codimension 2 foliations on simply connected compact 4-manifolds such that

  • Every leaf is diffeomorphic to $\mathbb R^2$
  • Every leaf is dense?

Same question for 5-manifolds and foliations of with leaves diffeomorphic to $\mathbb R^2$ or $\mathbb R^3$.

I am trying to show absence of Anosov diffeos on simply connected 4 and 5 manifolds and I would like to make sure I am not missing any important foliation theory background. So relevant references will be very much appreciated.

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  • $\begingroup$ Do you mean dimension 2 or codimension 2 ? Asking the leafs to be $\mathbb{R}^2$ tends to point toward the former. $\endgroup$
    – Benoît Kloeckner
    Sep 28, 2010 at 8:59
  • $\begingroup$ @Benoit: I believe in his context, what he would like to know is if there is some obstruction to having two transversal foliations (one of dimension and one of codimension 2) whose leaves are $\mathbb{R}^2$ and $\mathbb{R}^2$ (or $\mathbb{R}^3$) with that properties. $\endgroup$
    – rpotrie
    Sep 28, 2010 at 11:40
  • $\begingroup$ Sorry, I will edit the question. rpotrie: this would be the next question and I don't think this is possible, but probably very hard to show using topology only. $\endgroup$
    – Andrey Gogolev
    Sep 28, 2010 at 19:36
  • $\begingroup$ for holomorphic foliations of complex surfaces, there is a classification of Brunella: ams.org/mathscinet-getitem?mr=1474805 But you'd have to unravel the statement to see what it says in the simply-connected case. $\endgroup$
    – Ian Agol
    Sep 28, 2010 at 21:52

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