Searcharxiv⌕ Search

arXiv subjects

Sergey Maksymenko

Publications and source records attributed to Sergey Maksymenko.

8 recordsLinked to original sources
arXiv2015

Sections of Lie group actions and a theorem by M. Newman

Let $M$ be a smooth finite-dimensional manifold, $G$ be a Lie group, and $Φ:G \times M \to M$ be a smooth action. Consider the following mapping $ϕ: C^{\infty}(M,G) \to C^{\infty}(M,M)$, defined by $ϕ(α)(x) = α(x)\cdot x$, for $α\in C^{\infty}(M,G)$ and $x\in M$. In this paper we describe the structure of inverse images of elements of $C^{\infty}(M,M)$ under $ϕ$ for $\dim G=1$, i.e. when $G$ is either $\mathbb{R}$ or $S^1$. As an application we obtain a new proof of the well-known theorem by M. Newman concerning the interior of the fixed point set of a Lie group action.

math.DS↗
arXiv2015

Stabilizers and orbits of smooth functions

Let $f:R^m \to R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ such that $ϕ(0)=0$ the function $ϕ\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:\mathbb{R}^m \to \mathbb{R}^m$ such that $ϕ\circ f = f \circ h$ at $0\in \mathbb{R}^m$. The requirement is that $f$ belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated critical points, simple singularities and many others. We also globalize this result as follows. Let $M$ be a smooth compact manifold, $f:M \to [0,1]$ a surjective smooth function, $\mathrm{Diff}(M)$ the group of diffeomorphisms of $M$, and $\mathrm{Diff}^{[0,1]}(\mathbb{R})$ the group of diffeomorphisms of $\mathbb{R}$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $\mathrm{Diff}(M)$ and $\mathrm{Diff}(M) \times \mathrm{Diff}^{[0,1]}(\mathbb{R})$ on $C^{\infty}(M,R)$. Let $S_M(f)$, $S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we get the following homotopy equivalences $S_M(f) \approx S_{MR}(f)$ and $O_M \approx O_{MR}$. Similar results are obtained for smooth mappings $M \to S^1$.

math.FA↗
arXiv2015

Path-components of Morse mappings spaces of surfaces

Let $M$ be a compact surface and $P$ be a one dimensional manifold without boundary, that is the line $\mathbb{R}^1$ or a circle $S^1$. The classification of path-components of the space of Morse maps from $M$ into $P$ was recently obtained by S. V. Matveev and V. V. Sharko for the case $P=\mathbb{R}$. For $P=S^1$ the classification was obtained by the author. All this results can be reformulated as one theorem: "Two Morse maps $f,g:M \to P$ belong to the same path component of a space of Morse mappings from $M$ into $P$ if and only if they are homotopic and have the same number of crutucal points in each index and the same sets of positive and negative boundary circles". Here we give another independent proof of this theorem based on Lickorish's theorem on generators of homeotopy group of surface.

math.GT↗
arXiv2006

Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $π_k O(f)=π_k M$ for $k\geq 3$, and $π_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $π_1 O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$. We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.

math.GT↗
arXiv2005

Consecutive shifts along orbits of vector fields

Let $M$ be a smooth ($C^{\infty}$) manifold, $F_1,...,F_n$ be vector fields on $M$ generating the corresponding flows $Φ_1,...,Φ_n$, and $α_1,...,α_{n}:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$ by $$f(x)= Φ_n (... (Φ_2 (Φ_1 (x,α_1(x)), α_2(x)), ..., α_n(x)).$$ In this note we give a necessary and sufficient condition on vector fields $F_1,...,F_n$ and smooth functions $α_1,...,α_{n}$ for $f$ to be a local diffeomorphism. It turns out that this condition is invariant with respect to the simultaneous permutation of the corresponding vector fields and functions.

math.DG↗
arXiv2005

Stabilizers and orbits of circle-valued smooth functions

Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author proved that under mild conditions on $f\in C^{\infty}(M,R^1)$ the corresponding stabilizers and orbits are homotopy equivalent: $S_{MR} \sim S_{M}$ and $O_{MR} \sim O_M$. These results are extended here to the actions on $C^{\infty}(M,S^1)$. It is proved that under the similar conditions (that are rather typical) we have that $S_{MS}\sim S_M$ and $O_{MS} \sim O_M \times S^1$.

math.FA↗
arXiv2004

Smooth shifts along flows

Let $Φ$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(Φ)$ and $D(Φ)$ of $C^{\infty}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $Φ$. Let also $E_{0}(Φ)$ and $D_{0}(Φ)$ be the identity path components of $E(Φ)$ and $D(Φ)$ with compact-open topology. We prove that under mild conditions on fixed points of $Φ$ the inclusion $D_{0}(Φ) \subset E_{0}(Φ)$ is a homotopy equivalence and these spaces are either contractible or homotopically equivalent to the circle.

math.GT↗