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Almost contact 5-manifolds are contact. (English) Zbl 1333.53116

The main result proved in the paper: On a closed oriented 5-dimensional manifold, there exists a contact structure in every homotopy class of almost contact structures.
Using the techniques developed in this article, it is not possible to conclude anything about the number of distinct contact distributions that may occur in a given homotopy class of almost contact distributions. Thus, the obtained result states that there is at least one. In the article [F. Presas, Geom. Topol. 11, 2203–2225 (2007; Zbl 1132.57023)], one finds examples with more structures. It follows from the construction that the contact structure is PS-overtwisted [K. Niederkrüger, Algebr. Geom. Topol. 6, 2473–2508 (2006; Zbl 1129.53056)]; [K. Niederkrüger and F. Presas, Geom. Topol. 14, No. 2, 719–754 (2010; Zbl 1186.57020)] and therefore it is nonfillable.
The case of nonorientable 5-dimensional manifolds is also discussed.

MSC:

53D10 Contact manifolds (general theory)
53D15 Almost contact and almost symplectic manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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