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Loops in Reeb Graphs of n-Manifolds
  • Irina Gelbukh
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 1 June 2018
We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions. Expand
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 setExpand
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.Expand
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 $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$,Expand
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 hasExpand
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 aExpand
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 abelianExpand
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 ofExpand
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.
Isotropy index for the connected sum and the direct product of manifolds
A subspace or subgroup is isotropic under a bilinear map if the restriction of the map on it is trivial. We study maximal isotropic subspaces or subgroups under skew-symmetric maps, and in particularExpand