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Constructing symplectic forms on 4–manifolds which vanish on circles. (English) Zbl 1054.57027

Given a smooth, closed, oriented 4-manifold \(X\) and \(\alpha \in H_2(X,\mathbb{Z})\) such that \(\alpha.\alpha > 0\), a closed 2-form \(\omega\) is constructed, Poincaré dual to \(\alpha\), which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by \(d = (c_1(s)^2 -3\sigma(X) -2\chi(X))/4\), where \(s\) is a certain spin\(^\mathbb C\)-structure naturally associated to \(\omega\).

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57M50 General geometric structures on low-dimensional manifolds
32Q60 Almost complex manifolds
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