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Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized:

In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any compact, oriented manifold there is a bijection between the set of contact structures up to isotopy and the set of open book decomposition up to the operation of positive stabilization (meaning that any two stabilized manifolds give rise under Giroux's theorem, to the same contact structure up to isotopy). Open books are known to exist for all odd-dimensional manifolds.

Are there similar relationships on (2n+1)-manifolds, n>1 say compact and oriented, between contact structures and open books, or between open books and some other property?

Thanks.

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Giroux proved that for every contact manifold $(M^{2n+1},\alpha)$, there exists an open book supporting the contact form. An open book consists of a codimension two submanifold $K^{2n-1}\subseteq M^{2n+1}$ and a fibration $M\setminus K\to S^1$ (which is "standard" in some tubular neighborhood of $K$) with fibers $F_t$. Such an open book is said to support a contact form $\alpha$ if:

  • $\alpha$ restricts to a contact form on the binding $K^{2n-1}\subseteq M^{2n+1}$.
  • $d\alpha$ is a symplectic form on the pages $F_t$, and the associated Liouville vector field $X_\alpha$ on $F_t$ is outward pointing along $\partial F_t$ (equivalently, the orientation on $K$ induced by the contact form $\alpha$ agrees with the orientation on $K=\partial F_t$ induced by the symplectic form $d\alpha$ on $F_t$).

See http://arxiv.org/abs/math/0305129 for the proof. Conversely, any exact symplectomorphism $\varphi$ of a Liouville domain $(F,d\lambda)$, gives rise to a contact structure on the resulting open book with page $F$ and monodromy $\varphi$.

The question of whether there is a unique (up to positive stabilization) open book supporting a given contact structure is (as you mention) a theorem of Giroux in dimension three and is open in higher dimensions.

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  • $\begingroup$ Thanks, what do we mean by the binding in higher dimensions? For dimension 3 we have an $S^1$ -knot, what do we have for higher dimensions, some special embedding, maybe an isomorphic embedding of $S^{2n+1-2}=S^{2n-1}$ in $M^{2n+1}$? $\endgroup$ – Guest Dec 7 '14 at 1:29
  • $\begingroup$ See the edit above. The binding is just any codimension two submanifold, not necessarily a sphere. $\endgroup$ – John Pardon Dec 7 '14 at 1:39
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    $\begingroup$ For the record, one must point out that there are very important orientation assumptions to be added to the definition, see Giroux's paper. $\endgroup$ – Patrick Massot Jan 8 '15 at 10:22
  • $\begingroup$ Hi Patrick! I've edited the answer, and hopefully it is correct now. $\endgroup$ – John Pardon Jan 9 '15 at 5:20

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