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Contact structures on 1-connected 5-manifolds. (English) Zbl 0724.57017
A smooth \((2n+1)\)-dimensional manifold is said to be a contact manifold if it admits a differential 1-form \(\alpha\) such that \(\alpha \bigwedge (d\alpha)^n\) is nowhere zero. A 1-form with this property is called a contact form. The structure group of a contact manifold can be reduced to \(U(n)\times 1\); such a reduction is called an almost contact structure. A classical result of Lutz and Martinet states that orientable 3-manifolds admit a contact form in every homotopy class of almost contact structures. The author proves the corresponding result for simply- connected 5-manifolds, which were classified by D. Barden.
The proof uses recent results of A. Weinstein and D. McDuff. A. Weinstein [Hokkaido Math. J. 20, No. 2, 241–251 (1991; Zbl 0737.57012)] showed that under certain technical conditions it is possible to perform surgery on a contact manifold to obtain another contact manifold. Using this method, it can be shown that every simply-connected 5-manifold which admits an almost contact structure (for which the only obstruction is the third integral Stiefel-Whitney class) does in fact admit a contact form.
A contact form in every homotopy class of almost contact structures is obtained by lifting symplectic forms on \(B = \mathbb{CP}^2\#\overline{\mathbb{CP}^2\) to contact forms on \(S^1\)-bundles over \(B\) using the well known Boothby-Wang fibration. It was shown by D. McDuff [J. Am. Math. Soc. 3, No. 3, 679–712 (1990; Zbl 0723.53019)] that symplectic forms on \(B\) exist in abundance.
Reviewer: Hansjörg Geiges

MSC:
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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