# Prishlyak O. O.

### Deformations in the general position of the optimal functions on oriented surfaces with boundary

Hladysh B. I., Prishlyak O. O.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1028-1039

UDC 516.91

It is considered simple functions with non-degenerated singularities on smooth compact oriented surfaces with the boundary.
Authors describe a connection between optimality and polarity of Morse functions, $m$-functions and $mm$-functions on smooth compact oriented connected surfaces. The concept of an equipped Kronrod – Reeb graph is used to define a deformation in general position. Also, it is obtained the whole list of deformations of simple functions of one of abovedescribed class on torus, 2-dimensional disc with the boundary and on connected sum of two toruses.

### Functions with nondegenerate critical points on the boundary of the surface

Hladysh B. I., Prishlyak O. O.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 28-37

We prove an analog of the Morse theorem in the case where the critical point belongs to the boundary of an $n$-dimensional manifold and find the least number of critical points for the Morse functions defined on the surfaces whose critical points coincide with the critical points of their restriction to boundary.

### Equivalence of closed 1-forms on surfaces with edge

Budnyts'ka T. V., Prishlyak O. O.

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1455-1473

We investigate closed 1-forms with isolated zeros on surfaces with edge. A criterion for the topological equivalence of closed 1-forms is proved.

### Topological Classification of *m*-Fields on Two- and Three-Dimensional Manifolds with Boundary

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 799-805

We consider *m*-fields that are generalizations of the Morse–Smale vector fields for manifolds with boundary. We construct complete topological invariants of *m*-fields on surfaces and *m*-fields without closed trajectories on three-dimensional manifolds. We also prove criteria for the topological equivalence of *m*-fields.

### Topological Equivalence of Morse–Smale Vector Fields with beh2 on Three-Dimensional Manifolds

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 492-500

For the Morse–Smale vector fields with beh2 on three-dimensional manifolds, we construct complete topological invariants: diagram, minimal diagram, and recognizing graph. We prove a criterion for the topological equivalence of these vector fields.

### Conjugacy of Morse Functions on Surfaces with Values on a Straight Line and Circle

Ukr. Mat. Zh. - 2000. - 52, № 10. - pp. 1421-1425

We investigate the conjugacy of Morse functions on closed surfaces. By using cellular decompositions of surfaces, we formulate a criterion for the conjugacy of Morse functions. We establish a criterion for the conjugacy of mappings into a circle with nondegenerate critical points.

### Isomorphisms of combinatorial block decompositions of three-dimensional manifolds

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 568–571

For three-dimensional manifolds with the structure of a combinatorial block complex, we construct an invariant that allows one to verify the existence of isomorphisms, between these manifolds. For complexes of small dimensionality, we solve the problem on the possibility of extending the isomorphisms of subcomplexes to those of complexes.

### Vector fields with a given set of singular points

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1373–1384

Theorems on the existence of vector fields with given sets of indexes of isolated singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated singular point of the gradient field is not greater than one.

### New polynomials of knots

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1230–1235

For some knots and links with respect to regular isotopy, we introduce a new invariant, which is a Laurent polynomial in three variables. The properties of this invariant are studied.

### Minimal handle decomposition of smooth simply connected five-dimensional manifolds

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1714–1720

A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.

### Minimal Morse functions on a pair of manifolds

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 143-144

The existence theorem for a minimal Morse function on a pair of manifolds $(M_n,N_k)$, where $n - k ≥ 3,\; k ≥ 6$, is proved.