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The number of split points of a Morse form and the structure of its foliation. (English) Zbl 1324.57004
In the paper under review the author studies closed manifolds equipped with a closed smooth one-form, having Morse-type singularities. This means that locally in a neighborhood of a singular point, the one-form $$\omega$$ writes $$\omega =df$$ where $$f$$ is a non-degenerate quadratic form. Such a closed one-form defines a codimension one foliation outside its singular set, and some work has been done in the comprehension of the relation between the topology of the leaves of such a foliation and the combinatorics involving the arrangements of the singularities. Such singularities are originally divided into two classes: centers and saddles. In the paper under review the author introduces a class of so called split conic singularities, which are saddle-type singularities which are somehow accumulated by leaves coming from two distinct center-type singularities. The motivation is that when passing such a split point, a leaf splits into two in a sense made clear by the author. In the paper the author introduces and studies the number $$d(\omega)= \frac { a(\omega) - b(\omega)}{2} + 1$$, where $$a(\omega)$$ is the number of split points of $$\omega$$ and $$b(\omega)$$ is the number of centers of $$\omega$$. The author proves $$d(\omega)$$ is non-negative and shows some other inequalities associating $$d(\omega)$$ to some important characteristics of the foliation. Relations with Novikov’s theory of closed 1-forms and their singularities are then clear. The sharpness of the results is discussed. The paper can be quite useful for those interested in the field and related areas.

##### MSC:
 57R30 Foliations in differential topology; geometric theory
##### Keywords:
Morse form; singularities; foliation; foliation graph
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