E. A. Polulyakh

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Articles: 3

Foliations with all non-closed leaves on non-compact surfaces

Sergiy Maksymenko, Eugene Polulyakh

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 266-282

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.

On the theorem converse to Jordan's curve theorem

Eugene Polulyakh

Methods Funct. Anal. Topology 6 (2000), no. 4, 56-69

On imbedding of closed 2-dimensional disks into $ R^{2}$

E. A. Polulyakh

Methods Funct. Anal. Topology 4 (1998), no. 2, 76-94


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