Minimal Morse functions on a pair of manifolds

The existence theorem for a minimal Morse function on a pair of manifolds (M ⁿ,N ^k), wheren-k 3,k 6, is proved.Translated from Ukrainskii Matematiccheskii Zhurnal, Vol. 45, No. 1, pp. 143–144, January, 1993.     

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

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.     

Vector fields with a given set of singular points

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. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1373–1384, October, 1997.     

Isomorphisms of combinatorial block decompositions of three-dimensional manifolds

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. Shevchenko National University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal., Vol. 51, No. 4, pp. 568–571, April, 1999.     

Deformations in the General Position of the Optimal Functions on Oriented Surfaces with Boundary

Ukrainian Mathematical Journal - We consider simple functions with nondegenerate singularities on smooth compact oriented surfaces with boundary. The relationship between the optimality and...     

Morse--Smale Vector Fields without Closed Trajectories on $$3 $$ -Manifolds

We study Morse--Smale vector fields on $3$ -manifolds. A topological classification of such vector fields in terms of homeomorphisms of surfaces with two sets of circles on them is given. An algorithm verifying the existence of such homeomorphisms is constructed.     

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

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.     

New polynomials of knots

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. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1230–1235, September, 1997.     

Complete topological invariants of Morse-Smale flows and handle decompositions of 3-manifolds

We construct a topological invariant for the canonical decomposition on prime and round handles associated with a Morse-Smale flow on a closed 3-manifold. We prove that the flows are topologically equivalent if and only if their invariants coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 185–196, 2005.     

Minimal handle decomposition of smooth simply connected five-dimensional manifolds

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.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1714–1720, December, 1994.This research was partially supported by the International Scientific Foundation, grant No. V6F000.