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Singular points of a Morsian form and foliations. (English. Russian original) Zbl 0914.58006
Mosc. Univ. Math. Bull. 51, No. 4, 33-36 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 4, 37-40 (1996).
Let $$M$$ be a smooth compact connected orientable $$n$$-dimensional manifold on which the 1-form $$\omega$$ with Morse peculiarities is defined. On the manifold $$M$$ a foliation with peculiarities $$\mathcal F_\omega$$ is specified. The irrationality power of the form $$\omega$$ is determined by $\text{dirr } \omega = rk_Q\Biggl\{\int_{z_1} \omega,\dots, \int_{z_m} \omega \Biggr\} - 1,$ where $$z_1,\dots,z_m$$ is the basis in $$H_1(M)$$. It is proved that in the case of a compact foliation the irrationality power of the form and the number of homologically independent layers are determined by the correlation of the number of special points of index 0 and 1.

##### MSC:
 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 2.2e+71 Applications of Lie groups to the sciences; explicit representations
##### Keywords:
Morse forms; foliation; singular points