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Open books on contact five-manifolds. (English) Zbl 1143.53078
From the introduction: At the ICM of 2002 E. Giroux announced his results on the relation between manifolds and open book decompositions. The easy part of this results (and the part that we shall use) is a generalization of a construction due to W. P. Thurston and H. E. Winkelnkemper [Proc. Am. Math. Soc. 52, 345–347 (1975; Zbl 0312.53028)]; one can adapt certain open book decompositions to contact structures, thus giving a procedure to construct contact structures using open books. Roughly speaking Giroux’s construction goes as follows. Take a compact Stein manifold \(P\) or more generally an exact symplectic manifold with boundary and a symplecticomorphism \(\psi\) of \(P\) that is the identity near the boundary of \(P\). The mapping torus of \((P,\psi)\) can be shown to admit a natural contact structure. On the other hand a neighborhood of the binding \(\partial P\times D^2\) has a natural contact structure that can be glued to the contact structure on the mapping torus, therefore giving rise to a closed contact manifold with an adapted contact structure.
In this paper, we will use Giroux’s construction to reprove a theorem on the existence of contact structures on five-manifolds due to H. Geiges [Mathematika 38, No. 2, 303–311 (1991; Zbl 0724.57017)]. More precisely, we shall prove the following theorem (Geiges): Let \(M\) be a simply-connected five manifold. Then \(M\) admits a contact structure in every homotopy class of almost contact structure. The main idea of our alternative proof is very simple. Using the classification of simply-connected five-manifolds, we can reduce the problem to finding contact structures on certain model manifolds. We do this by explicit construction using Giroux’s procedure. Although this is not necessary in the construction of Giroux, we will always take Stein surfaces as pages. Since the classification of simply-connected five-manifolds is determined by the homology groups and the second Stiefel-Whitney class, it is sufficient to track these topological invariants.
Reviewer: Ioan Pop (Iaşi)

MSC:
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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