• Publications
  • Influence
Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds
For a connected locally path-connected topological space X and a continuous function f on it such that its Reeb graph Rf is a finite topological graph, we show that the cycle rank of Rf, i.e., the
The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
Abstract We study b1′ $b_{1}'$ (M), the co-rank of the fundamental group of a smooth closed connected manifold M. We calculate this value for the direct product of manifolds. We characterize the set
Loops in Reeb Graphs of n-Manifolds
TLDR
The set of possible values of the number of loops in the Reeb graph is described in terms of the co-rank of the fundamental group of the manifold and it is shown that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions.
On the structure of a Morse form foliation
The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated.
Close cohomologous Morse forms with compact leaves
We study the topology of foliations of close cohomologous Morse forms (smooth closed 1-forms with non-degenerate singularities) on a smooth closed oriented manifold. We show that if a closed form has
Co-rank and Betti number of a group
For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian
The number of split points of a Morse form and the structure of its foliation
Sharp bounds are given that connect split points — conic singularities of a special type — of a Morse form with the global structure of its foliation.
On compact leaves of a Morse form foliation
On a compact oriented manifold without boundary, we consider a closed 1-form with singularities of Morse type, called Morse form. We give criteria for the foliation de ned by this form to have a
The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface
On a closed orientable surface M 2 of genus g, we consider the foliation of a weakly generic Morse form ! on M 2 and show that for such forms c(!) + m(!) = g 1 k(!), wh ere c(!) is the number of
Number of minimal components and homologically independent compact leaves for a morse form foliation
The numbers m ( ω ) of minimal components and c ( ω ) of homologically independent compact leaves of the foliation of a Morse form ω on a connected smooth closed oriented manifold M are studied in
...
...