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Sergiy Maksymenko

Publications and source records attributed to Sergiy Maksymenko.

At least 37 records · Page 2Linked to original sources
arXiv2020

Actions of groups of foliated homeomorphisms on spaces of leaves

Let $Δ$ be a foliation on a topological manifold $X$, $Y$ be the space of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the factor topology with respect to $p$. Then the group $\mathcal{H}(X, Δ)$ of foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts on the space of leaves $Y$, which gives a homomorphism $ψ: \mathcal{H}(X, Δ) \to \mathcal{H}(Y)$. We present sufficient conditions when $ψ$ is continuous with respect to the corresponding compact open topologies. In fact similar results hold not only for foliations but for a more general class of partitions $Δ$ of locally compact Hausdorff spaces $X$.

math.GT↗
arXiv2020

Deformations of functions on surfaces by isotopic to the identity diffeomorphisms

Let $M$ be a compact surface and $P$ be either $\mathbb{R}$ or $S^1$. For a smooth map $f:M\to P$ and a closed subset $V\subset M$, denote by $\mathcal{S}(f,V)$ the group of diffeomorphisms $h$ of $M$ preserving $f$, i.e. satisfying the relation $f\circ h = f$, and fixed on $V$. Let also $\mathcal{S}'(f,V)$ be its subgroup consisting of diffeomorphisms isotopic relatively $V$ to the identity map $\mathrm{id}_{M}$ via isotopies that are not necessarily $f$-preserving. The groups $π_0 \mathcal{S}(f,V)$ and $π_0 \mathcal{S}'(f,V)$ can be regarded as analogues of mapping class group for $f$-preserving diffeomorphisms. The paper describes precise algebraic structure of groups $π_0 \mathcal{S}'(f,V)$ and some of their subgroups and quotients for a large class of smooth maps $f:M\to P$ containing all Morse maps, where $M$ is orientable and distinct from $2$-sphere and $2$-torus. In particular, it is shown that for certain subsets $V$ "adapted" in some sense with $f$, the groups $π_0 \mathcal{S}'(f,V)$ are solvable and Bieberbach.

math.GT↗
arXiv2019

Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points

Let $f:S^2\to \mathbb{R}$ be a Morse function on the $2$-sphere and $K$ be a connected component of some level set of $f$ containing at least one saddle critical point. Then $K$ is a $1$-dimensional CW-complex cellularly embedded into $S^2$, so the complement $S^2\setminus K$ is a union of open $2$-disks $D_1,\ldots, D_k$. Let $\mathcal{S}_{K}(f)$ be the group of isotopic to the identity diffeomorphisms of $S^2$ leaving invariant $K$ and also each level set $f^{-1}(c)$, $c\in\mathbb{R}$. Then each $h\in \mathcal{S}_{K}(f)$ induces a certain permutation $σ_{h}$ of those disks. Denote by $G = \{ σ_h \mid h \in \mathcal{S}_{K}(f)\}$ be the group of all such permutations. We prove that $G$ is isomorphic to a finite subgroup of $SO(3)$.

math.GT↗
arXiv2019

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $Γ_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by $\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$ induces a homeomorphism of $Γ_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)= \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $Γ_f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{O}_{f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this situation $Γ_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$ consists of more than one point.

math.GT↗
arXiv2019

Homotopy properties of smooth functions on the Möbius band

Let $B$ be a Möbius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $π_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on non-orientable surfaces $N$.

math.GT↗
arXiv2018

Automorphisms of Kronrod-Reeb graphs of Morse functions on compact surfaces

Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of difeomorphisms of $M$ isotopic to the identity. Denote by $\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms "preserving" $f$, i.e. the stabilizer of $f$ with respect to the right action of $\mathcal{D}_{\mathrm{id}}(M)$ on the space $\mathcal{C}^{\infty}(M,\mathbb{R})$ of smooth functions on $M$. Let also $\mathbf{G}(f)$ be the group of automorphisms of the Kronrod-Reeb graph of $f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is an important ingredient in determining the homotopy type of the orbit of $f$ with respect to the above action of $\mathcal{D}_{\mathrm{id}}(M)$ and it is trivial if $f$ is "generic", i.e. has at most one critical point at each level set $f^{-1}(c)$, $c\in\mathbb{R}$. For the case when $M$ is distinct from $2$-sphere and $2$-torus we present a precise description of the family $\mathbf{G}(M,\mathbb{R})$ of isomorphism classes of groups $\mathbf{G}(f)$, where $f$ runs over all Morse functions on $M$, and of its subfamily $\mathbf{G}^{smp}(M,\mathbb{R}) \subset \mathbf{G}(M,\mathbb{R})$ consisting of groups corresponding to simple Morse functions, i.e. functions having at most one critical point at each connected component of each level set. In fact, $\mathbf{G}(M,\mathbb{R})$, (resp. $\mathbf{G}^{smp}(M,\mathbb{R})$), coincides with the minimal family of isomorphism classes of groups containing the trivial group and closed with respect to direct products and also with respect to wreath products "from the top" with arbitrary finite cyclic groups, (resp. with group $\mathbb{Z}_2$ only).

math.GT↗
arXiv2017

Characterization of striped surfaces

Let $Z$ be a non-compact two-dimensional manifold and $Δ$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $Δ$ and each leaf of $Δ$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,Δ)$ glued from open strips $\mathbb{R}\times(0,1)$ with boundary leaves along some of their boundary intervals.

math.GT↗
arXiv2017

Homeotopy groups of one-dimensional foliations on surfaces

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $Δ$ on all of $Z$. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "striped" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{R}^2 \to \mathbb{R}$ without singularities. Previously, the first two authors studied the homotopy type of the group $\mathcal{H}(Δ)$ of homeomorphisms of $Z$ sending leaves of $Δ$ onto leaves, and shown that except for two cases the identity path component $\mathcal{H}_{0}(Δ)$ of $\mathcal{H}(Δ)$ is contractible. The aim of the present paper is to show that the quotient $\mathcal{H}(Δ)/ \mathcal{H}_{0}(Δ)$ can be identified with the group of automorphisms of a certain graph with additional structure encoding the "combinatorics" of gluing.

math.GT↗
arXiv2017

Symplectomorphisms of surfaces preserving a smooth function, I

Let $M$ be a compact orientable surface equipped with a volume form $ω$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $ω$. Let also $\mathcal{Z}_ω(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,ω)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $ω$ and $f$. We construct a canonical map $φ: \mathcal{Z}_ω(f) \to \mathcal{S}_{\mathrm{id}}(f,ω)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. In particular, we obtain that $\mathcal{S}_{\mathrm{id}}(f,ω)$ is either contractible or homotopy equivalent to the circle. Similar results hold in fact for a larger class of maps $M\to P$ whose singularities are equivalent to homogeneous polynomials without multiple factors.

math.SG↗
arXiv2017

Contractibility of manifolds by means of stochastic flows

In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $π_1 M=0$. We prove that in fact under the similar conditions the manifold $M$ is contractible, that is all homotopy groups $π_k M$, $k\geq1$, vanish.

math.PR↗
arXiv2016

One-dimensional foliations on topological manifolds

Let $X$ be an $(n+1)$-dimensional manifold, $Δ$ be a one-dimensional foliation on $X$, and $p: X \to X / Δ$ be a quotient map. We will say that a leaf $ω$ of $Δ$ is special whenever the space of leaves $X / Δ$ is not Hausdorff at $ω$. We present necessary and sufficient conditions for the map $p: X \to X / Δ$ to be a locally trivial fibration under assumptions that all leaves of $Δ$ are non-compact and the family of all special leaves of $Δ$ is locally finite.

math.GT↗
arXiv2016

Foliations with all non-closed leaves on non-compact surfaces

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $Δ$ be a foliation on $X$ such that each leaf $ω\inΔ$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $Δ$.

math.GT↗
arXiv2016

Topological stability of continuous functions with respect to averaging by measures with locally constant densities

Let $μ$ be a measure on $[-1,1]$. Then for every continuous function $f:\mathbb{R}\to\mathbb{R}$ and $α>0$ one can define its averaging $f_α:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_α(x) = \int_{-1}^{1} f(x+tα)dμ. \] In arXiv:1509.06064 the authors studied the problem when $f_α$ is topologically equivalent to $f$ for all $α>0$ and call this property a topological stability of $f$ under averagings with respect to measure $μ$. Similarly one can define topological stability of a germ of $f$ at some point $x\in\mathbb{R}$. It was shown that for a continuous function $f:\mathbb{R}\to\mathbb{R}$ having only finitely many local extremes and any measure $μ$ topological stability of averagings of germs of $f$ at local extremes implies topological stability of averagings of $f$. In the present paper we prove the converse statement: topological stability of averagings of $f$ implies topological stability of averagings of its germs at local extremes. Moreover, in arXiv:1509.06064 it was also extablished a sufficient condition for topological stability of averagings of local extremes with respect to measures with finite supports. In the present paper we obtain similar sufficient conditions for measures with locally continuous and in particular with locally constant densities.

math.CA↗
arXiv2015

Hamiltonian vector fields of homogeneous polynomials in two variables

Let $g:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty}$ diffeomorphisms $(\mathbb{R}^2,O)\to(\mathbb{R}^2,O)$ that preserve orbits of $G$. Let also $E_{\mathrm{id}}(G,O)$ be the identity component of $E(G,O)$ with respect to $C^1$-topology. Suppose that $g$ has no multiple prime factors. Then we prove that for every $h\in E_{\mathrm{id}}(G,O)$ there exists a germ of a smooth function $α:\mathbb{R}^2\to\mathbb{R}$ at $O$ such that $h(z)=G_{α(z)}(z)$.

math.DS↗
arXiv2015

Homotopy dimension of orbits of Morse functions on surfaces

Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the connected components of $O$ have the homotopy type of a finite-dimensional CW-complex. Actually, these connected components are homotopy equivalent to a certain covering space of the $n$-th configuration space of the interior of $M$. As a consequence we obtain that the fundamental group of $O$ is a subgroup of the $n$-th braid group of $M$.

math.AT↗
arXiv2015

Connected components of partition preserving diffeomorphisms

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) = \cdots = S(f,1)$ for all such $f$ and that $S(f,1) \not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.

math.DS↗
arXiv2015

Local inverses of shift maps along orbits of flows

Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article ["Smooth shifts along trajectories of flows", Topol. Appl. 130 (2003) 183-204, arXiv:math/0106199] concerning the homotopy types of the group of diffeomorphisms preserving orbits of $F$ contains two errors. They imply that the principal statement of that paper holds under additional assumptions on $F$. Unfortunately this result was essentially used in another paper of mine ["Homotopy types of stabilizers and orbits of Morse functions on surfaces" Ann. Glob. Anal. Geom., 29 no. 3, (2006) 241-285, arXiv:math/0310067]. The aim of this article is to expose the results of the first paper in a right way, extend them to a larger class of flows with degenerate singularities, and show that the results of the second paper remain true.

math.DS↗