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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
##### Keywords:
foliation; four-manifold; almost-complex
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