Ph.D. (mathematics)
Hide detailsA finite graph is homeomorphic to the Reeb graph of a Morse–Bott function. Mathematica Slovaca, 71(3):757–772, 2021; doi: 10.1515/ms-2021-0018.
Abstract: 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 $n\ge 2$. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number $k\ge 3$ of critical values, and in a few special cases with $k<3$. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion in ${\mathrm{\mathbb{R}}}^{3}$.
Key words: Reeb graph, Morse–Bott function, immersion, height function, surface
MSC 2010: Primary 58C05, 58K65; Secondary 68U05, 05C60
JCR impact factor: 0.654, Q3 (2019)
This site provides a preprint version of the paper. Private camera-ready version, final version, and first page.
Morse–Bott functions with two critical values on a surface. Czechoslovak Mathematical Journal, 71(3):865–880, 2021; doi: 10.21136/CMJ.2021.0125-20.
Abstract: 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 functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with non-orientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse–Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.
Key words: Morse–Bott function, height function, surface, critical values, Reeb graph
MSC 2020: 58C05, 57K20, 05C38
JCR impact factor: 0.412, Q4 (2019)
Pre-print version. Private camera-ready version and final version.
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. Filomat, 33(7):2031–2049, 2019; doi: 10.2298/FIL1907031G.
Abstract: 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 ${b}_{1}\left({R}_{f}\right)$, in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group ${\pi}_{1}\left(X\right)$, the condition of local path-connectedness being important since generally ${b}_{1}\left({R}_{f}\right)$ can even exceed ${b}_{1}\left(X\right)$. We give some practical methods for calculating the co-rank of ${\pi}_{1}\left(X\right)$ and a closely related value, the isotropy index. We apply our bound to improve upper bounds on the distortion of the Reeb quotient map, and thus on the Gromov-Hausdorff approximation of the space by Reeb graphs, for the distance function on a compact geodesic space and for a simple Morse function on a closed Riemannian manifold. This distortion is bounded from below by what we call the Reeb width $b\left(M\right)$ of a metric space $M$, which guarantees that any real-valued continuous function on $M$ has large enough contour (connected component of a level set). We show that for a Riemannian manifold, $b\left(M\right)$ is non-zero and give a lower bound on it in terms of characteristics of the manifold. In particular, we show that any real-valued continuous function on a closed Euclidean unit ball $E$ of dimension at least two has a contour $C$ with $diam(C\cap \partial E)\ge \sqrt{3}$.
Key words: Reeb graph, co-rank of the fundamental group, metric distortion, Gromov-Hausdorff distance, Riemannian manifolds
MSC 2010: Primary 05C38; Secondary 58C05, 68U05
JCR impact factor: 0.789 (2018)
This site provides a preprint version of the paper.
Compact and locally dense leaves of a closed one-form foliation. Journal of Mathematical Analysis and Applications, 464:1275–1289, 2018; doi: 10.1016/j.jmaa.2018.04.053.
Abstract: 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 notion of minimal components, previously used in the context of Morse form foliations, to general foliations. Finally, we give a condition for the form foliation to have only closed leaves (closed in the complement of the singular set).
Key words: closed one-form, foliation, compact leaves, locally dense leaves, minimal components
MSC 2010: 57R30, 58K65
JCR impact factor: 1.064 (2016)
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Loops in Reeb graphs of n-manifolds. Discrete & Computational Geometry, 59(4):843–863, 2018; doi: 10.1007/s00454-017-9957-9.
Abstract: 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 of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface.
Key words: Reeb graph, contour tree, number of loops, Morse function, co-rank of the fundamental group
MSC 2010: 05C38, 05E45, 58C05
JCR impact factor: 0.724 (2016)
Sufficient conditions for the compactifiability of a closed one-form foliation. Turkish Journal of Mathematics, 41:1344–1353, 2017; doi: 10.3906/mat-1602-95.
Abstract: 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 conditions, the foliation can be defined by closed 1-forms with the ranks of their groups of periods in a certain range. In addition, we describe the structure of the group generated by the homology classes of all compact leaves of the foliation.
Key words: Closed one-form foliation, compact leaves, form's rank.
MSC 2010: 57R30, 58A10, 53C65, 58K65
JCR impact factor: 0.378 (2015)
Isotropy index for the connected sum and the direct product of manifolds. Publicationes Mathematicae Debrecen, 90(3–4):287–310, 2017; doi: 10.5486/PMD.2017.7409.
Abstract: 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 M, we describe the geometric meaning of the isotropic subgroups of the first cohomology group with different coefficients under the cup product. We calculate the corresponding isotropy index, as well as the set of ranks of all maximal isotropic subgroups, for the connected sum and the direct product of manifolds. Finally, we study the relationship of the isotropy index with the first Betti number and the co-rank of the fundamental group. We also discuss applications of these results to the topology of foliations.
Key words: Isotropic subspace, cohomology, cup product
MSC: 15A63, 15A03, 58K65
JCR impact factor: 0.431 (2016)
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The co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliations. Mathematica Slovaca, 67(3):645–656, 2017; Zbl 06738450; doi: 10.1515/ms-2016-0298.
Abstract: We study ${b}_{1}^{\prime}\left(M\right)$, 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 ${b}_{1}^{\prime}\left(M\right)$ and the first Betti number ${b}_{1}\left(M\right)$ by explicitly constructing manifolds with any possible combination of ${b}_{1}^{\prime}\left(M\right)$ and ${b}_{1}\left(M\right)$ in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold $M$ and a Morse form $\omega $ on it for any possible combination of ${b}_{1}^{\prime}\left(M\right)$, ${b}_{1}\left(M\right)$, $m\left(\omega \right)$, and $c\left(\omega \right)$, where $m\left(\omega \right)$ is the number of minimal components and $c\left(\omega \right)$ is the maximum number of homologically independent compact leaves of $\omega $.
Key words: co-rank, inner rank, manifold, fundamental group, direct product, Morse form foliation
MSC 2010: 14F35, 57N65, 57R30
JCR impact factor: 0.346 (2016)
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Co-rank and Betti number of a group. Czechoslovak Mathematical Journal, 65(2):565–567, 2015; Zbl 1363.20034; doi: 10.1007/s10587-015-0195-0.
Abstract: 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 some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive with respect to the free product and the direct product of groups. Our results are important for the theory of foliations and for manifold topology, where the corresponding notions are related with the cut-number (or genus) and the isotropy index of the manifold, as well as with the operations of connected sum and direct product of manifolds.
Key words: co-rank, inner rank, fundamental group
MSC 2010: 20E05, 20F34, 14F35
JCR impact factor: 0.284 (2015)
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The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface. Rocky Mountain Journal of Mathematics, 43(5):1537–1552, 2013; Zbl 1280.57021; doi: 10.1216/RMJ-2013-43-5-1537.
Abstract: On a closed orientable surface ${M}_{g}^{2}$ of genus $g$, we consider the foliation of a weakly generic Morse form $\omega $ on ${M}_{g}^{2}$ and show that for such forms $c\left(\omega \right)+m\left(\omega \right)=g-\genfrac{}{}{0.1ex}{}{1}{2}k\left(\omega \right)$, where $c\left(\omega \right)$ is the number of homologically independent compact leaves of the foliation, $m\left(\omega \right)$ the number of its minimal components, and $k\left(\omega \right)$ the total number of singularities of $\omega $ surrounded by a minimal component. We also give lower bounds on $m\left(\omega \right)$ in terms of $k\left(\omega \right)$ and the form rank $rk\phantom{\rule{0.167em}{0ex}}\omega $ or the structure of $ker\phantom{\rule{0.167em}{0ex}}\left[\omega \right]$, where $\left[\omega \right]$ is the integration map.
Key words: Surface; Morse form foliation; number of minimal components
MSC 2000: 57R30, 58K65
JCR impact factor: 0.491 (2013)
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Close cohomologous Morse forms with compact leaves. Czechoslovak Mathematical Journal, 63(2):515–528, 2013; Zbl 1289.57009; doi: 10.1007/s10587-013-0034-0.
Abstract: We study the topology of foliations of close cohomologous Morse forms on a smooth closed oriented manifold. We show that if a closed form has a compact leave $\gamma $, then any close cohomologous form has a compact leave close to $\gamma $. Then we prove that the set of Morse forms with compactifiable foliations is open in a cohomology class, and the number of homologically independent compact leaves does not decrease under small perturbation of the form; moreover, for generic forms this number is locally constant.
Key words: Morse form foliation, compact leaves, cohomology class
MSC 2010: 57R30, 58K65
JCR impact factor: 0.300 (2012)
This site provides a preprint version of the paper. See (probably open) final version on the journal's site. Private copy.
The number of split points of a Morse form and the structure of its foliation. Mathematica Slovaca, 63(2):331–348, 2013; Zbl 1324.57004; doi: 10.2478/s12175-013-0101-x.
Abstract: 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.
Key words: Morse form; singularities; foliation; foliation graph
MSC 2000: 57R30, 58K65
JCR impact factor: 0.451 (2013)
This site provides a preprint version of the paper. Private copy.
Structure of a Morse form foliation on a closed surface in terms of genus. Differential Geometry and its Applications, 29(4):473–492, 2011; Zbl 1223.57022; doi: 10.1016/j.difgeo.2011.04.029.
Erratum: In Corollary 25, "non-planar a compact singular leaf" should read "a non-planar compact singular leaf."
Abstract: We study the geometry of compact singular leaves $\gamma $ and minimal components ${C}^{\mathrm{min}}$ of the foliation $F$ of a Morse form $\omega $ on ${M}_{g}^{2}$ in terms of genus $g(\cdot )$. We show that $c\left(\omega \right)+\sum _{\gamma}g\left(V\right(\gamma \left)\right)+g(\bigcup \stackrel{\u203e}{{C}^{\mathrm{min}}})=g$, where $c\left(\omega \right)$ is the number of homologically independent compact leaves and $V(\cdot )$ is a small closed tubular neighborhood. This allows us to prove a criterion for compactness of the singular foliation $\stackrel{\u203e}{F}$, to estimate the number of its minimal components, and to give an upper bound on the rank $rk\omega $, in terms of genus.
Key words: Morse form foliation; minimal component; compact singular leave; genus; isotropic subgroup
MSC 2000: 57R30, 58K65
JCR impact factor: 0.669 (2009), 0.484 (2012)
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On collinear closed one-forms. Bulletin of the Australian Mathematical Society, 84(2):322–336, 2011; Zbl 1226.57040; doi: 10.1017/S0004972711002310.
Abstract: 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 existence of a compact leaf and to study homological properties of compact leaves.
Key words: differential one-form; singular set; foliation; compact leaves; cup-product
MSC 2000: 57R30, 58A10
JCR impact factor: 0.480 (2012)
This site provides a preprint version. See the paper on the journal's site and on the editor's site. Private copy.
On compact leaves of a Morse form foliation. Publicationes Mathematicae Debrecen, 78(1):37–48, 2011; Zbl 1240.57011; doi: 10.5486/PMD.2011.4369.
Abstract: 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.
Key words: Morse form foliation, compact leaves, collinear 1-forms, form rank
MSC 2000: 57R30, 58K65
JCR impact factor: 0.646 (2009), 0.568 (2010), 0.322 (2012)
This site provides a preprint version of the paper. Version "in print". Final version on the journal's site. Private copy.
Ranks of collinear Morse forms. Journal of Geometry and Physics, 61(2):425–435, 2011; Zbl 1210.57027; doi: 10.1016/j.geomphys.2010.10.010.
Erratum: Section 1, paragraph 3: "in general relativity and quantum cosmology" should read "in general relativity".
Abstract: 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 how it is related to the structure of the foliation and the manifold.
Key words: Collinear Morse forms, Morse form foliation, form's rank, minimal components, compact leaves
MSC 2000: 57R30, 58K65
JCR impact factor: 1.055 (2012)
This site provides a preprint version. Final version on the journal's website. Private copy.
Number of minimal components and homologically independent compact leaves for a Morse form foliation. Studia Scientiarum Mathematicarum Hungarica, 46(4):547–557, 2009; Zbl 1274.57005; doi: 10.1556/SScMath.2009.1108.
Erratum: page 551, "the next ${k}_{1}$ factors correspond to the set of weakly complete minimal components," should read "<...> not weakly complete <...>".
Abstract: The numbers $m\left(\omega \right)$ of minimal components and $c\left(\omega \right)$ of homologically independent compact leaves of the foliation of a Morse form $\omega $ on a connected smooth closed oriented manifold $M$ are studied in terms of the first non-commutative Betti number ${b}_{1}^{\prime}$. A sharp estimate $0\le m\left(\omega \right)+c\left(\omega \right)\le {b}_{1}^{\prime}$ is given. It is shown that all values of $m\left(\omega \right)+c\left(\omega \right)$, and in some cases all combinations of $m\left(\omega \right)$ and $c\left(\omega \right)$ with this condition, are reached on a given $M$. The corresponding issues are also studied in the classes of generic forms and compactifiable foliations.
Key words: Morse form foliation, minimal components, compact leaves
MSC 2000: 57R30, 58K65
JCR impact factor: 0.229 (2009), 0.421 (2012)
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On the structure of a Morse form foliation. Czechoslovak Mathematical Journal, 59(1):207–220, 2009; Zbl 1224.57010; doi: 10.1007/s10587-009-0015-5.
Abstract: The foliation of a Morse form $\omega $ 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 $\omega $. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of $rk\phantom{\rule{0.167em}{0ex}}\omega $ and $Sing\phantom{\rule{0.167em}{0ex}}\omega $. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if $\omega $ has more centers than conic singularities then ${b}_{1}\left(M\right)=0$ and thus the foliation has no minimal components and homologically non-trivial compact leaves, its foliation graph being a tree.
Key words: number of minimal components, number of maximal components, compact leaves, foliation graph, rank of a form
MSC 2000: 57R30, 58K65
JCR impact factor: 0.306 (2009), 0.300 (2012)
This site provides a preprint version. A final version on the journal's site (direct link to PDF), another final version on the journal's site. Private copy.
Presence of minimal components in a Morse form foliation. Differential Geometry and its Applications, 22(2):189–198, 2005; Zbl 1070.57016; doi: 10.1016/j.difgeo.2004.10.006.
Abstract: Conditions and a criterion for the presence of minimal components in the foliation of a Morse form $\omega $ on a smooth closed oriented manifold $M$ are given in terms of (1) the maximum rank of a subgroup in ${H}^{1}(M,\mathbb{Z})$ with trivial cup-product, (2) $ker\phantom{\rule{0.167em}{0ex}}\left[\omega \right]$, and (3) $rk\phantom{\rule{0.167em}{0ex}}\omega =rk\phantom{\rule{0.167em}{0ex}}im\left[\omega \right]$, where $\left[\omega \right]$ is the integration map.
Key words: Morse form foliation, minimal components, form rank, cup-product
MSC 2000: 57R30, 58K65
JCR impact factor: 0.391 (2005), 0.669 (2009), 0.484 (2012)
This site provides a draft version. Private copy.
Maximal isotropic subspaces of skew-symmetric bilinear mapping.
Moscow University Mathematics Bulletin
54(4):1–3,
1999;
Zbl 0957.57018.
Translated from:
Максимальные изотропные подпространства кососимметрического билинейного
отображения.
Вестник МГУ, Сер. матем., мех. 4:3–5, 1999.
Erratum: The paper deals with compactifiable foliations and not compact.
Zentralblatt review: The paper continues the author’s investigations [I. A. Mel’nikova, Math. Notes 58, No. 6, 1302-1305 (1995); translation from Mat. Zametki 58, No. 6, 872-877 (1995; Zbl 0857.57030); Russ. Math. Surv. 50, No. 2, 444-445 (1995); translation from Usp. Mat. Nauk 50, No. 3, 217-218 (1995; Zbl 0859.58005)] in which the compactness problem for the Morse form foliation on a closed manifold ${M}^{n}$ is considered. The author discusses the problem of calculation of the maximal isotropic subgroup in ${H}_{n-1}\left(M\right)$ with respect to the operation of intersection of homology classes. The upper and lower estimates are established and some examples are considered when $M={T}^{n}$ is an $n$-dimensional torus and $M={M}_{g}^{2}$.
MSC: 57R30, 57M07, 54H10 (according to Zentralblatt)
Full name of the journal of the original (Russian) publication: Vestnik Moskovskogo Universiteta Seriya 1 Matematika Mekhanika, 0579-9368.
JCR impact factor: 0.01 (1999)
Private copy. Private copy missing for English version.
Noncompact leaves of foliations of
Morse forms.
Mathematical Notes 63(6):760–763,
June 1998;
Zbl 0917.57022;
doi: 10.1007/BF02312769.
Translated from:
Некомпактные слои слоения морсовской формы.
Математические заметки 63(6):862–865, 1998.
Erratum: The paper deals with (non-)compactifiable leaves and not (non-)compact.
Abstract: 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.
Keywords: Morse forms, noncompact leaves of foliations, two-dimensional manifolds
Zentralblatt review: Let $M$ be a compact connected oriented manifold of dimension $n$ with a closed 1-form $\omega $ having only Morse singularities (Morse form). Let ${F}_{\omega}$ be a foliation with singularities on $M$ and $\left[\gamma \right]$ the homology class of a nonsingular compact leaf $\gamma \in {F}_{\omega}$. The image of the set of nonsingular compact leaves generates a subgroup ${H}_{\omega}$ in ${H}_{n-1}\left(M\right)$. By ${\Omega}_{i}$ denote the set of singular points of index $i$. In this note an inequality involving the number $s$ 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 $\omega $ is obtained. Theorem. The following inequality holds: ${r}_{k}{H}_{\omega}+s\le \genfrac{}{}{0.1ex}{}{1}{2}\left(\right|{\Omega}_{1}|-|{\Omega}_{0}\left|\right)+1$. P. Arnoux and G. Levitt [Invent. Math. 84, 141–156 (1986; Zbl 0561.58024)] obtained an estimate of $s$ in terms of characteristic of $M$: $s\le \genfrac{}{}{0.1ex}{}{1}{2}{b}_{1}\left(M\right)$. These two estimates coincide for $n=2$, and they are independent in the case $n>2$. The method is based on some results of graph theory.
Keywords: singular points of a foliation (according to Zentralblatt)
MSC: 57R30, 53C12, 57R20 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.344 (2010)
English: Download or preview this paper from Springer's site (RoMEO green publication according to ResearchGate). Private copy.
Russian: info and full text on the journal's site (open access). Private copy.
Properties of Morse forms that determine
compact foliations on ${M}_{g}^{2}$.
Mathematical Notes 60(6):714–716,
1996;
Zbl 0898.57012;
doi: 10.1007/BF02305168.
Translated from:
Свойства морсовской формы, определяющей компактное слоение на ${M}_{g}^{2}$.
Математические заметки 60(6):942–945, 1996.
Erratum: The paper deals with compactifiable foliations and not compact.
Abstract: 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 $\left[\omega \right]:{H}_{1}\left(M\right)\to \mathbb{R}$. 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 $Ker\left[\omega \right]\subset {H}_{1}\left({M}_{g}^{2}\right)$ with the topology of the foliation. We consider the structure of the subgroup $Ker\left[\omega \right]$ for a compact foliation and prove a criterion for the compactness of a foliation.
Keywords: two-dimensional manifold, foliation, Morse form, integration over cycles.
Zentralblatt review: P. Arnoux and G. Levitt [Invent. Math. 84, 141–156 (1986; Zbl 0561.58024); ibid. 88, 635–667 (1987; Zbl 0594.57014)] 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 $\left[\omega \right]:{H}_{1}\left(M\right)\to \mathbb{R}$. In this paper, the author considers the foliation of a Morse form on a two-dimensional manifold ${M}_{g}^{2}$. He studies the relationship of the subgroup $Ker\left[\omega \right]\subset {H}_{1}\left({M}_{g}^{2}\right)$ with the topology of the foliation, considers the structure of the subgroup $Ker\left[\omega \right]$ for a compact foliation and proves a criterion for the compactness of a foliation.
Keywords: integration over cycles; Morse form; two-dimensional manifold; compact foliation (according to Zentralblatt)
MSC: 57R30, 58C25, 58K99 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.344 (2010)
English: Download or preview this paper from Springer's site (RoMEO green publication according to ResearchGate). Private copy.
Russian: info and full text on the journal's site (open access). Private copy.
Singular points of a Morsian form and foliations.
Moscow University Mathematics Bulletin
51(4):33–36,
1996;
Zbl 0914.58006.
Translated from:
Особые точки морсовской формы и слоения.
Вестник МГУ, Сер. матем., мех. 4:37–40,
1996.
Errata: The title should have been translated as "Singularities of a Morse form and foliation". Where the paper mentions the irrationality degree $dirr$, it would be more common to refer to the form rank: $dirr\phantom{\rule{0.167em}{0ex}}\omega =rk\phantom{\rule{0.167em}{0ex}}\omega -1$. The paper deals with compactifiable foliations and not compact foliations.
Zentralblatt review (with corrected translation of terminology): Let $M$ be a smooth compact connected orientable $n$-dimensional manifold on which a 1-form $\omega $ with Morse singularities is defined. On the manifold $M$, a foliation with singularities ${F}_{\omega}$ is defined. The irrationality degree of the form $\omega $ is determined by $dirr\phantom{\rule{0.167em}{0ex}}\omega ={rk}_{\mathbb{Q}}\{\underset{{z}_{1}}{\int}\omega ,...,\underset{{z}_{m}}{\int}\omega \}-1$, where ${z}_{1},...,{z}_{m}$ is the basis in ${H}_{1}\left(M\right)$. It is proved that in the case of a compact foliation, the irrationality degree of the form and the number of homologically independent leaves are determined by the difference of the numbers of singularities of index 0 and 1.
Keywords: Morse forms; foliation; singular points (according to Zentralblatt)
MSC 2000: 58E05 22E70 (according to Zentralblatt)
JCR impact factor: 0.01 (1999)
Full name of the journal of the original (Russian) publication: Vestnik Moskovskogo Universiteta Seriya 1 Matematika Mekhanika, 0579-9368.
Private copy missing for English version. Private copy.
A test for compactness of a
foliation.
Mathematical Notes 58(6):1302–1305, 1995;
Zbl 0857.57030;
doi: 10.1007/BF02304889.
Translated from:
Признак компактности слоения.
Математические заметки 58(6):872–877, 1995.
Erratum: The paper deals with compactifiable foliations and not compact.
Abstract: 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.
Zentralblatt review: We investigate foliations on smooth manifolds that are determined by a closed 1-form with Morse singularities. The problem of investigating the topological structure of level surfaces for such a form was posed by S. P. Novikov [Russ. Math. Surv. 37, No. 5, 1-56 (1982); translation from Usp. Mat. Nauk 37, No. 5(227), 3-49 (1982; Zbl 0571.58011)]. This problem was treated in [S. P. Novikov, Tr. Mat. Inst. Steklova 166, 201-209 (1984; Zbl 0553.58005); A. V. Zorich, Math. USSR, Izv. 31, No. 3, 635-655 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 6, 1322-1344 (1987; Zbl 0668.58004); Le Ty Kuok Tkhang, Math. Notes 44, No. 1/2, 556-562 (1988); translation from Mat. Zametki 44, No. 1, 124-133 (1988; Zbl 0656.58007); L. A. Alaniya, Russ. Math. Surv. 46, No. 3, 211-212 (1991); translation from Usp. Mat. Nauk 46, No. 3(279), 179-180 (1991; Zbl 0736.58001)]. The present paper is devoted to the compactness problem for level surfaces. We introduce the notion of degree of compactness and prove a test for compactness expressed in the terms of the degree. The present paper is a natural continuation of [the author, Math. Notes 53, No. 3, 356-358 (1993); translation from Mat. Zametki 53, No. 3, 158-160 (1993; Zbl 0809.57018)].
MSC: 57R30 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.344 (2010)
English: Download or preview this paper from Springer's site (RoMEO green publication according to ResearchGate). Private copy.
Russian: info and full text on the journal's site (open access). Private copy.
A test for non-compactness of the foliation of a Morse
form.
Russian Mathematical Surveys 50(2):444–445, 1995;
Zbl 0859.58005;
doi: 10.1070/RM1995v050n02ABEH002092.
Translated from:
Признак некомпактности слоения морсовской формы.
Успехи математических наук 50(3):217–218, 1995.
Erratum: The paper deals with (non-)compactifiable foliations and not (non-)compact.
The paper has no abstract.
MSC: 57R30, 53C12, 58Kxx
Zentralblatt review: 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-compactness of a foliation is a case of general position.
Keywords: Morse form; foliations; non-compactness (according to Zentralblatt)
MSC: 58C25, 58K99, 58E05 (according to Zentralblatt)
JCR impact factor: 0.363 (2001), 0.496 (2010)
English: Download or preview this paper from IOP's site (RoMEO classification unknown to ResearchGate). Private copy.
Russian: info and full text on the journal's site (open access). Private copy.
An indicator of the noncompactness of a foliation on ${M}_{g}^{2}$.
Mathematical Notes 53(3):356–358, 1993;
Zbl 0809.57018;
doi: 10.1007/BF01207728.
Translated from:
Признак некомпактности слоения на
${M}_{g}^{2}$.
Математические заметки 53(3):158–160, 1993.
Erratum: The paper deals with (non-)compactifiable foliations and not (non-)compact.
The paper has no abstract.
Zentralblatt review: Let $\omega $ be a closed form on a manifold $M$ and possessing nondegenerate isolated singularities. A point $p\in M$ is called a regular singularity of $\omega $, if in some neighbourhood $O\left(p\right)$ $\omega =df$, where $f$ is a Morse function having a singularity at $p$. The form $\omega $ determines a foliation ${F}_{\omega}$ on the set $M-Sing\phantom{\rule{0.167em}{0ex}}\omega $. Let $M={M}_{g}^{2}$, the orientable closed surface of genus 2. The homology classes $\left[\gamma \right]$ of the nonsingular compact leaves of ${F}_{\omega}$ generate a subgroup of ${H}_{1}\left({M}_{g}^{2}\right)$ denoted by ${H}_{\omega}$. If $\left[{z}_{1}\right],...,\left[{z}_{2g}\right]$ is a basis of ${H}_{1}\left({M}_{g}^{2}\right)$ we define $dirr\phantom{\rule{0.167em}{0ex}}\omega ={rk}_{\mathbb{Q}}\{\underset{{z}_{1}}{\int}\omega ,...,\underset{{z}_{2g}}{\int}\omega \}-1$. By ${M}_{\omega}$ is denoted the set obtained by discarding all maximal neighbourhoods consisting of diffeomorphic compact leaves and all leaves which can be compactified by adding singular points. Theorem 1. ${M}_{\omega}=\varnothing $ ⇔ $rk{H}_{\omega}=g$. Theorem 2. If $\omega $ is a closed form with Morse singularities given on ${M}_{g}^{2}$ ($g\ne 0$) such that $dirr\phantom{\rule{0.167em}{0ex}}\omega \ge g$, then the foliation ${F}_{\omega}$ has a noncompact fiber.
JCR impact factor: 0.157 (2001), 0.344 (2010)
English: Download or preview this paper (English) from Springer's site (RoMEO green publication according to ResearchGate). Private copy.
Russian: info and full text on the journal's site (open access). Private copy.
Realization of a graph as the Reeb graph of a Morse, Morse–Bott or round function. International Online Conference "Algebraic and Geometric Methods of Analysis" dedicated to the memory of Yuriy Trokhymchuk (17.03.1928–18.12.2019), Odessa, Ukraine, May 25–28, 2021.
Topology of the Reeb graph. Howard Mathematics Colloquium, Howard University, Washington DC, March 26, 2021.
Abstract: 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 dynamical systems. It also finds practical applications in computer graphics, shape analysis, machine learning, big data analysis, and geometric model databases. We give a criterion for the Reeb graph to have the structure of a finite graph (generally it is not: we give counterexamples) and describe general properties of such graphs. We also consider the realization problem: the conditions for a finite graph to be the Reeb graph of a function of some class, such as any smooth function, Morse, Morse–Bott, or round function.
Keynote talk. 3rd International Conference on Innovations in Computing, CGC College of Engineering, Mohali, Punjab, India, 12–13 December 2019.
Rank of a maximal subgroup in ${H}^{1}(M,\mathbb{Z})$ with trivial cup-product. Fourth International Conference of Applied Mathematics and Computing (FICAMC), Plovdiv, Bulgaria, August 12–18, 2007.
Abstract: Let $M$ be a smooth closed oriented manifold, ${h}^{\mathrm{max}}\left(M\right)$ be the maximal rank of a maximal subgroup in ${H}^{1}(M,\mathbb{Z})$ with trivial cup-product, and ${h}^{\mathrm{min}}\left(M\right)$ the minimal rank of such a subgroup. It has been shown that the value of $h\left(M\right)$ characterizes the topology of Morse form foliations on $M$: e.g., if $rk\phantom{\rule{0.167em}{0ex}}\omega >h\left(M\right)$, where $\omega $ is a Morse form on $M$, then its foliation has a minimal component. We give upper and lower bounds on ${h}^{\mathrm{max}}\left(M\right)$ and ${h}^{\mathrm{min}}\left(M\right)$ in terms of the first and second Betti numbers. In addition, we calculate these values for a connected sum and direct product of manifolds.
Only abstract was published.
Locally mirrored copy of the proceedings.
Compact foliations of a Morse form on ${M}_{g}^{2}$. International Conference on Topology and Applications in memory of P.S. Alexandroff (1896–1982). Moscow, Russia, May 27–31, 1996.
Erratum: The talk deals with compactifiable foliations and not compact.
This talk had no publication.
Compact foliations of Morse
forms.
PhD thesis (in Russian). MSU, 1995.
И.А. Мельникова.
Компактные слоения морсовских форм.
Дисс. ... к.ф.-м.н., МГУ, 1995.
Erratum: The thesis deals with compactifiable foliations and not compact.
Summary: 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.
See the officially published abstract.
Last modified 2021-03-28 |