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37
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Introduction
I work in topology and foliation theory. In particular, I study foliations of closed one-forms and Morse forms on orientable manifolds, as well as Reeb graphs of arbitrary functions and on arbitrary topological spaces.
My personal page is www.i.gelbukh.com.
Skills and Expertise
Publications
Publications (37)
We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse-Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the c...
We prove criteria for a graph to be the Reeb graph of a function of
a given class on a closed manifold: Morse–Bott, round, and in general smooth
functions whose critical set consists of a finite number of submanifolds. The criteria
are given in terms of whether the graph admits an orientation, which we call S-
good orientation, with certain conditi...
We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $R^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We a...
![a A graph G. b Blocks Bi\documentclass[12pt]{minimal}...](publication/358213556/figure/fig4/AS:1148928204513281@1650937238104/a-A-graphG-b-BlocksBidocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
Given a set S of integers, for a directed acyclic multigraph, we say
that it has an S-good orientation if all its sources and sinks have degrees in S; in
these terms, the existing notion of good orientation is a {1}-good orientation.
We give a criterion for a pseudograph to admit an S-good orientation in terms
of the structure of its leaf blocks. T...
The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classif...
In 2021, Saeki proved that if a smooth function f on a closed manifold M has a finite number of critical values, then its Reeb graph R f is a finite graph. We show by examples that in general case, the Reeb graph is not necessarily a finite graph and can have very quaint topology, e.g., it can be not 1-dimensional or even non-Hausdorff. Generalizin...
We show that any non-trivial finite connected graph (allowing loop edges and multiple edges) is isomorphic to the Reeb graph of a Morse circle-valued function on a closed n-manifold of a given dimension n ≥ 2; this manifold roughly resembles a thick version of the graph, we present its construction and study its properties. In the case of surfaces...
We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g., as fiber bundles over already constructed manifolds with the same property). We study properties of such fu...
We give criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse, Morse--Bott, round, and in general a smooth function whose critical set consists of a finite number of submanifolds. Unlike previous works on this topic, these criteria are given not in terms of the graph orientation, but in terms of its st...
Reeb graph of a function is a space obtained by contracting the connected components
of the level sets of the function to points, endowed with the quotient topology (plus an
additional structure in the case of a smooth function). This notion is useful in topological
classification of functions and, under the name of Lyapunov graph, in theory of dyn...
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 $R(f)$, i.e., the first Betti number $b_1(R_ f)$, in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental gro...
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 $R_f$, i.e., the first Betti number $b_1(R_f)$, in computational geometry called \emph{number of loops}, is bounded from above by the co-rank of the fundamental...
The Reeb graph of a smooth function on a connected smooth closed orientable $n$-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank...
We study a foliation defined by a possibly singular smooth closed one-form on a connected smooth closed orientable manifold. We prove two bounds on the total number of homologically independent compact leaves and of connected components of the union of all locally dense leaves, which we call minimal components. In particular, we generalize the noti...
![Figure 1. Two maximal subgroups of different ranks: ⟨[N ]⟩ and ⟨[T1],...](profile/Irina-Gelbukh-2/publication/312027185/figure/fig1/AS:544268246843392@1506775070584/Two-maximal-subgroups-of-different-ranks-N-and-T1-T2-The-sides-of-each-cube_Q320.jpg)
We study the foliation defined by a closed 1-form on a connected smooth closed orientable manifold. We call such a foliation compactifiable if all its leaves are closed in the complement of the singular set. In this paper, we give sufficient conditions for compactifiability of the foliation in homological terms. We also show that under these condit...
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 particular the isotropy index---the maximum dimension of an isotropic subspace or maximum rank of an isotropic subgroup. For a smooth closed orientable manifold...
We study $b'_1(M)$, 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 of all possible combinations of $b'_1(M)$
and the first Betti number $b_1(M)$ by explicitly constructing manifolds with
any possible combination of $b'_1(M)$ and $b_...
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 quotient, called the Betti number. We show that any combination of the group’s rank, co-rank, and Betti number within obvious constraints is realized for so...
On a closed orientable surface M2g of genus g, we consider the foliation of a weakly generic Morse form ω on M2g and show that for such forms c(ω) + m(ω) = g - (1/2)k(ω), where c(ω) is the number of homologically independent compact leaves of the foliation, m(ω) is the number of its minimal components, and k(ω) is the total number of singularities...
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 a compact leave γ, then any close cohomologous form has a compact leave close to γ. Then we prove that the set of Morse forms with compactifiable foli...
Sharp bounds are given that connect split points—conic sin-gularities of a special type—of a Morse form with the global structure of its foliation.
We study one-forms with zero wedge-product, which we call collinear, and their foliations. We characterise the set of forms that define a given foliation, with special attention to closed forms and forms with small singular sets. We apply the notion of collinearity to give a criterion for the existence of a compact leaf and to study homological pro...
We study the geometry of compact singular leaves γ and minimal components Cmin of the foliation Fω of a Morse form ω on a genus g closed surface Mg2 in terms of genus g(⁎). We show that c(ω)+∑γg(V(γ))+g(⋃Cmin¯)=g, where c(ω) is the number of homologically independent compact leaves and V(⁎) is a small closed tubular neighborhood. This allows us to...
On a smooth closed n-manifold, we consider Morse forms with wedge-product zero; we call such forms collinear. This is an equivalence relation. Collinearity classes are classified by the underlying foliation; so, in other words, we study the set of Morse forms that define the same foliation. We describe the set of the ranks of such forms and show ho...
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 defined by this form to have a compact leaf, to have k homologically independent compact leaves, and to have no minimal components.
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 terms of the first non-commutative Betti number b 1 ' (M). A sharp estimate 0≦m(ω)+c(ω)≦b 1 ' (M) is given. It is shown that all values of m(ω)+c(ω), and in so...
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. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components an...
Let M be a smooth closed oriented manifold, h(M)h max (M) be the maximal rank of a maximal subgroup in H 1 (M,Z) with trivial cup-product, and h min (M) the minimal rank of such a subgroup. It has been shown that the value of h(M) characterizes the topology of Morse form foliations on M: e.g., if rkω > h(M), where ω is a Morse form on M, then its f...
Conditions and a criterion for the presence of minimal components in the foliation of a Morse form ω on a smooth closed oriented manifold M are given in terms of (1) the maximum rank of a subgroup in H1(M,Z) with trivial cup-product, (2) ker[ω], and (3) rkω=defrkim[ω], where [ω] is the integration map.
In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.
In [1, 2] P. Arnoux and G. Levitt showed that the topology of the foliation of a Morse form $\omega$ on a compact manifold is closely related to the structure of the integration mapping $[\omega]: H_1(M) \to R$. In this paper we consider the foliation of a Morse form on a two-dimensional manifold $M_g^2$. We study the relationship of the subgroup $...
The author studies foliations determined by a closed 1-form with Morse singularities on smooth compact manifolds. More precisely, the author investigates the problem of the existence of a non-compact leaf, verifies a test for non-compactness of a foliation in terms of the degree of irrationality of the considered 1-form, and shows that the non-comp...
We investigate foliations on smooth manifolds that are determined by a closed 1-form with Morse singularities. We introduce the notion of the degree of compactness and prove a test for compactness.
Sufficient condition for compactifiability of a Morse form foliation, an upper bound on the rank of a Morse form defining a compactifiable foliation, and a lower bound on the number of the conic singularities of a Morse form defining a compactifiable foliation are given.
Questions
Question (1)
Closed 1-forms are well studied in foliation topology, algebraic geometry, and theory of manifolds. What are their applications in physics?
Projects
Projects (4)
Morse-Bott functions is a natural generalization of Morse functions. Their importance for the theory of Reeb graphs is due to the fact that every finite graph is homeomorphic to the Reeb graph of a Morse-Bott function. Our goal is to give criteria for a finite graph to be isomorphic to the Reeb graph of a Morse-Bott function, a Morse or a round function, and to study properties of these functions using their Reeb graphs.
Study of (infinite) Reeb graphs of functions with any critical set or on any topological space. Generally, Reeb graph is not a graph, not Hausdorff, not one-dimensional. Existing studies are restricted to functions on manifolds with some conditions of finiteness on the critical set, thus, to finite Reeb graphs.
A Reeb graph of a function is the space of connected components of its level sets, endowed with the quotient topology. Reeb graphs are widely used in computational geometry, computer graphics, geometric model databases, data visualization, and for shape analysis in computational topology.
Study of closed one-form foliations in terms of the form (singular set, integration map, the form's rank) and the manifold (isotropy index, co-rank of the fundamental group). Such foliations appear in quantum gravity, M-theory, relativity.
On a closed orientable manifold, the form defines a foliation of the complement to the set of its singularities. We study conditions for all leaves to be closed, the number of classes of closed and locally dense leaves, the set of the forms defining a given foliations, and related topics. Many relevant properties of foliations of Morse forms (locally the differential of a Morse function), which are open and dense in the space of closed one-forms, generalize to this case.
