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Existence of foliations on 4-manifolds. (English) Zbl 1052.57040
Let \(M\) be an oriented 4-manifold and \(c\in H^2(M;\mathbb Z)\) an integral lift of \(w_2(M)\) with a splitting \(c=\tau+\nu\). Set \(m=\frac{1}{4}(p_1(M)+2\chi(M)-c^2)\) and \(n=\frac{1}{4}(-p_1(M)+2\chi(M)+c^2-4\tau\nu)\). If \(m, n\geq 0\) then there is a singular foliation \(\mathcal F\) with \(e(T_\mathcal F)=\tau\), \(e(N_\mathcal F)=\nu\) and \(m+n\) singularities. The singularities may be modelled on the levels of the complex functions \((z_1,z_2)\mapsto z_1/z_2\) or \((z_1,z_2)\mapsto z_1z_2\), with \(n\) of them for complex coordinates respecting orientation and \(m\) of them for complex coordinates reversing orientation. Conditions under which given embedded surfaces can be transverse to the foliation or be leaves of the foliation are considered.
MSC:
57R30 Foliations in differential topology; geometric theory
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
32Q60 Almost complex manifolds
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