Geiges, Hansjörg Contact structures on 1-connected 5-manifolds. (English) Zbl 0724.57017 Mathematika 38, No. 2, 303-311 (1991). 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 Cited in 1 ReviewCited in 17 Documents MSC: 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:Legendre embeddings; contact manifold; almost contact structure; surgery; 5-manifold; symplectic forms; Boothby-Wang fibration Citations:Zbl 0737.57012; Zbl 0723.53019 PDFBibTeX XMLCite \textit{H. Geiges}, Mathematika 38, No. 2, 303--311 (1991; Zbl 0724.57017) Full Text: DOI References: [1] DOI: 10.2307/1970165 · Zbl 0084.39204 · doi:10.2307/1970165 [2] DOI: 10.2307/1970192 · Zbl 0092.39301 · doi:10.2307/1970192 [3] Weinstein, Hokkaido Math. J. 20 pp 241– (1991) · Zbl 0737.57012 · doi:10.14492/hokmj/1381413841 [4] DOI: 10.1016/0040-9383(67)90020-1 · Zbl 0173.26102 · doi:10.1016/0040-9383(67)90020-1 [5] Thomas, Banach Center Publ 18 pp 255– (1986) [6] Thomas, Mathematika 24 pp 237– (1977) [7] Meckert, Ann. tnst. Fourier 32 pp 251– (1982) · Zbl 0471.58001 · doi:10.5802/aif.888 [8] DOI: 10.2307/1990934 · Zbl 0723.53019 · doi:10.2307/1990934 [9] DOI: 10.1090/S0002-9904-1961-10690-3 · Zbl 0192.29601 · doi:10.1090/S0002-9904-1961-10690-3 [10] Lutz, C. R. Acad. Sc. Paris 282A pp 591– (1976) [11] DOI: 10.2307/2372928 · Zbl 0124.16302 · doi:10.2307/2372928 [12] DOI: 10.1007/BFb0068901 · doi:10.1007/BFb0068901 [13] DOI: 10.1007/BF02566892 · Zbl 0102.38603 · doi:10.1007/BF02566892 [14] Gromov, Partial Differential Relations (1986) · doi:10.1007/978-3-662-02267-2 [15] DOI: 10.2307/1970702 · Zbl 0136.20602 · doi:10.2307/1970702 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.