arXiv subjects

Sergiy Maksymenko

Publications and source records attributed to Sergiy Maksymenko.

At least 19 recordsLinked to original sources

arXiv2026

One-dimensional non-Hausdorff manifolds and CW complexes

Say that a connected non-Hausdorff one-dimensional manifold $M$ is "graph-like", whenever the set $M_{br}$ of its non-Hausdorff (called "branch") points is locally finite and every connected component of its complement has a countable base. We prove that for every graph-like manifold $M$ there exists a one-dimensional CW complex $K$ with the set of vertices $K^{(0)}$, a vertex $v\in K^{(0)}$ and a (surjective) quotient map $π\colon M \to K\setminus\{v\}$ such that $π^{-1}(K^{(0)}\setminus \{v\}) = M_{br}\cup\partial M$. Conversely, existence of such quotient map $π$ and the assumption that $M_{br}$ is nowhere dense imply that $M$ is graph-like. Moreover, in this case $K\setminus\{v\}$ is the minimal Hausdorff factor of $M$, that is, for every continuous map $f\colon M \to N$ into a Hausdorff space $N$ there exists a unique continuous map $\hat{f}\colon K\setminus\{v\}\to N$ such that $f = \hat{f}\circπ$.

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arXiv2025

Smooth functions that split a Klein bottle into two Möbius bands

Given a compact surface $M$, consider the right action $\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$, $(f, h) \mapsto f\circ h$, of the group $\mathcal{D}(M)$ of $\mathcal{C}^{\infty}$ diffeomorphisms of $M$ on the space $\mathcal{C}^{\infty}(M)$ of $\mathcal{C}^{\infty}$ functions on $M$. For $f\in\mathcal{C}^{\infty}(M)$ denote by $\mathcal{O}(f)$ its orbit, and by $\mathcal{O}_f(f)$ the path component of $\mathcal{O}(f)$ containing $f$. The paper continues a series of computations by many authors of homotopy types of orbits $\mathcal{O}_f(f)$ of smooth functions on compact surfaces. We provide here the computations of $\mathcal{O}_f(f)$ for a special class of functions $f\in\mathcal{C}^{\infty}(K)$ on the Klein bottle $K$ having the following properties: (i) at each critical point $f$ is smoothly equivalent to some homogeneous polynomial (e.g. $f$ is Morse), and (ii) there is a regular connected component $α$ of a level set of $f$ such that $K\setminusα$ is a disjoint union of two open Möbius bands, with closures $M_1$ and $M_2$. Let $f_i = f|_{M_i}$ be the restriction of $f$ to the Möbius band $M_i$, $i=1,2$, and $\mathcal{O}_{f_i}(f_i)$ be the path component of $f_i$ in its orbit with respect to the above action of $\mathcal{D}(M_i)$. The possible homotopy types of $\mathcal{O}_{f_i}(f_i)$ are explicitly computed earlier. We prove that $\mathcal{O}_f(f)$ is homotopy equivalent to $\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$.

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arXiv2025

Differentiable structures on a union of two open sets

In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$. Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures. We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.

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arXiv2025

Deformational symmetries of smooth functions on non-orientable surfaces

Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f\circ h$ for $f\in \mathcal{C}^{\infty}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$. Denote by $\mathcal{F}(M)$ the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisting of function $f:M\to\mathbb{R}$ taking constant values on connected components of $\partial{M}$, having no critical points on $\partial{M}$, and such that at each of its critical points $z$ the function $f$ is $\mathcal{C}^{\infty}$ equivalent to some homogenenous polynomial without multiple factors. In particular, $\mathcal{F}(M)$ contains all Morse maps. Let also $\mathcal{O}(f) = \{ f\circ h \mid h\in\mathcal{D}(M) \}$ be the orbit of $f$. Previously it was computed the algebraic structure of $π_1\mathcal{O}(f)$ for all $f\in\mathcal{F}(M)$, where $M$ is any orientable compact surface distinct from $2$-sphere. In the present paper we compute the group $π_0\mathcal{S}(f,\partial\mathbb{M})$, where $\mathbb{M}$ is a Möbius band, and $\mathcal{S}(f,\partial\mathbb{M}) = \{ h\in\mathcal{D}(\mathbb{M}) \mid f\circ h = f, \ h|_{\partial \mathbb{M}} = \mathrm{id}_{\mathbb{M}}\}$ is the subgroup of the corresponding stabilizer of $f$ consisting of diffeomorphisms fixed on the boundary $\partial \mathbb{M}$. As a consequence we obtain an explicit algebraic description of $π_1\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein bottle and projective plane.

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arXiv2024

Vector bundle automorphisms preserving Morse-Bott foliations

Let $M$ be a smooth manifold and $\mathcal{F}$ a Morse-Bott foliation on $M$ with a compact critical manifold $Σ$. Denote by $\mathcal{D}(\mathcal{F})$ the group of diffeomorphisms of $M$ leaving invariant each leaf of $\mathcal{F}$. Under certain assumptions on $\mathcal{F}$ it is shown that the computation of the homotopy type of $\mathcal{D}(\mathcal{F})$ reduces to three rather independent groups: the group of diffeomorphisms of $Σ$, the group of vector bundle automorphisms of some regular neighborhood of $Σ$, and the subgroup of $\mathcal{D}(\mathcal{F})$ consisting of diffeomorphisms fixed near $Σ$. Examples of computations of homotopy types of groups $\mathcal{D}(\mathcal{F})$ for such foliations are also presented.

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arXiv2024

Classification of differentiable structures on the non-Hausdorff line with two origins

We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in\mathbb{N}\cup\{\infty\}$ let $H$ be the group of homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of $h$ to $\mathbb{R}\setminus0$ is a $\mathcal{C}^{k}$-diffeomorphism. Let also $D$ be the subgroup of $H$ consisting of $\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{R}$ also fixing $0$. It is shown that there is a natural bijection between $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism fixing both origins) and double $D$-coset classes $D \setminus H / D = \{ D h D \mid h \in H\}$. Moreover, the set of all $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double $(D,\pm)$-coset classes $D \setminus H^{\pm} / D = \{ D h D \cup D h^{-1} D \mid h \in H\}$. In particular, in contrast with the real line, the line with two origins $\mathbb{L}$ admits uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures for each $k=1,2,\ldots,\infty$.

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arXiv2024

Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2

Let $\mathcal{F}$ be a Morse-Bott foliation on the solid torus $T=S^1\times D^2$ into $2$-tori parallel to the boundary and one singular central circle. Gluing two copies of $T$ by some diffeomorphism between their boundaries, one gets a lens space $L_{p,q}$ with a Morse-Bott foliation $\mathcal{F}_{p,q}$ obtained from $\mathcal{F}$ on each copy of $T$ and thus consisting of two singluar circles and parallel $2$-tori. In the previous paper [O. Khokliuk, S. Maksymenko, Journ. Homot. Rel. Struct., 2024, 18, 313-356] there were computed weak homotopy types of the groups $\mathcal{D}^{lp}(\mathcal{F}_{p,q})$ of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group $\mathcal{D}_{+}^{fol}(\mathcal{F}_{p,q})$ of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences.

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arXiv2024

Topological actions of wreath products

Let $G$ and $H$ be two groups acting on path connected topological spaces $X$ and $Y$ respectively. Assume that $H$ is finite of order $m$ and the quotient maps $p:X\to X/G$ and $q:Y\to Y/H$ are regular coverings. Then it is well-known that the wreath product $G\wr H$ naturally acts on $W = X^m\times Y$, so that the quotient map $r:W \to W/(G\wr H)$ is also a regular covering. We give an explicit description of $π_1(W/(G\wr H))$ as a certain wreath product $π_1(X/G)\,\wr_{\partial_Y}π_1(Y/H)$ corresponding to a non-effective action of $π_1(Y/H)$ on the set of maps $H\toπ_1(X/G)$ via the boundary homomorphism $\partial_{Y}:π_1(Y/H) \to H$ of the covering map $q$. Such a statement is known and usually exploited only when $X$ and $Y$ are contractible, in which case $W$ is also contractible, and thus $W/(G\wr H)$ is the classifying space of $G\wr H$. The applications are given to the computation of the homotopy types of orbits of typical smooth functions $f$ on orientable compact surfaces $M$ with respect to the natural right action of the groups $\mathcal{D}(M)$ of diffeomorphisms of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$.

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arXiv2023

Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1

Let $T= S^1\times D^2$ be the solid torus, $\mathcal{F}$ the Morse-Bott foliation on $T$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$, which is the central circle of the torus $T$, and $\mathcal{D}(\mathcal{F},\partial T)$ the group of diffeomorphisms of $T$ fixed on $\partial T$ and leaving each leaf of the foliation $\mathcal{F}$ invariant. We prove that $\mathcal{D}(\mathcal{F},\partial T)$ is contractible. Gluing two copies of $T$ by some diffeomorphism between their boundaries, we will get a lens space $L_{p,q}$ with a Morse-Bott foliation $\mathcal{F}_{p,q}$ obtained from $\mathcal{F}$ on each copy of $T$. We also compute the homotopy type of the group $\mathcal{D}(\mathcal{F}_{p,q})$ of diffeomorphisms of $L_{p,q}$ leaving invariant each leaf of $\mathcal{F}_{p,q}$.

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arXiv2023

Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliations $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$.

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arXiv2022

Foliated and compactly supported isotopies of regular neighborhoods

Let $\mathcal{F}$ be a foliation with a "singular" submanifold $B$ on a smooth manifold $M$ and $p:E \to B$ be a regular neighborhood of $B$ in $M$. Under certain "homogeneity" assumptions on $\mathcal{F}$ near $B$ we prove that every leaf preserving diffeomorphism $h$ of $M$ is isotopic via a leaf preserving isotopy to a diffeomorphism which coincides with some vector bundle morphism of $E$ near $B$. This result is mutually a foliated and compactly supported variant of a well known statement that every diffeomorphism $h$ of $\mathbb{R}^n$ fixing the origin is isotopic to the linear isomorphism induced by its Jacobi matrix of $h$ at $0$. We also present applications to the computations of the homotopy type of the group of leaf preserving diffeomorphisms of $\mathcal{F}$.

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arXiv2022

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

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 some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals which gives a foliation $Δ$ on all of $Z$. Denote by $\mathcal{H}(Z,Δ)$ the group of all homeomorphisms of $Z$ that maps leaves of $Δ$ onto leaves and by $\mathcal{H}(Z/Δ)$ the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group $π_0\mathcal{H}(Z,Δ)$ with a group of automorphisms of a certain graph $G$ with the additional structure which encodes the combinatorics of gluing $Z$ from strips. That graph is in a certain sense dual to the space of leaves $Z/Δ$. On the other hand, for every $h\in\mathcal{H}(Z,Δ)$ the induced permutation $k$ of leaves of $Δ$ is in fact a homeomorphism of $Z/Δ$ and the correspondence $h\mapsto k$ is a homomorphism $ψ:\mathcal{H}(Δ)\to\mathcal{H}(Z/Δ)$. The aim of the present paper is to show that $ψ$ induces a homomorphism of the corresponding homeotopy groups $ψ_0:π_0\mathcal{H}(Z,Δ)\toπ_0\mathcal{H}(Z/Δ)$ which turns out to be either injective or having a kernel $\mathbb{Z}_2$. This gives a dual description of $π_0\mathcal{H}(Z,Δ)$ in terms of the space of leaves.

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arXiv2021

Fundamental groupoids and homotopy types of non-compact surfaces

The paper contains an application of van Kampen theorem for groupoids for computation of homotopy types of certain class of non-compact foliated surfaces obtained by gluing at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.

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arXiv2021

Morse index of saddle equilibria of gradient-like flows on connected sums of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

math.DS↗

arXiv2021

Deformations of functions on surfaces

The paper contains a review on recent progress in the deformational properties of smooth maps from compact surfaces $M$ to a one-dimensional manifold $P$. It covers description of homotopy types of stabilizers and orbits of a large class of smooth functions on surfaces obtained by the author, E. Kudryavtseva, B. Feshchenko, I. Kuznietsova, Yu. Soroka, A. Kravchenko. We also present here a new direct proof of the fact that for generic Morse maps the connected components their orbits are homotopy equivalent to finite products of circles.

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arXiv2021

Reversing orientation homeomorphisms of surfaces

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. is a mirror symmetry with respect to some line. The obtained results hold in fact for a larger class of maps with isolated singularities from connected compact orientable surfaces to the real line and the circle.

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arXiv2020

Smooth approximations and their applications to homotopy types

Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0\leq r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence. It is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $Ω(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

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arXiv2020

Diffeomorphisms preserving Morse-Bott functions

Let $f:M\to\mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set $Σ_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f) = \{h \in \mathcal{D}(M) \mid f\circ h=h \}$ the group of diffeomorphisms of $M$ preserving $f$ and let $\mathcal{D}(Σ_f)$ be the group of diffeomorphisms of $Σ_f$. We prove that the "restriction to $Σ_f$" map $ρ:\mathcal{S}(f) \to \mathcal{D}(Σ_f)$, $ρ(h) = h|_{Σ_f}$, is a locally trivial fibration over its image $ρ(\mathcal{S}(f))$.

math.DG↗