Description

• Title:
  Deformational symmetries of smooth functions on non-orientable surfaces
• Creator:
  Kuznietsova, Iryna;
  Maksymenko, Sergiy
• Description:
  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 functions $\func\colon M\to\mathbb{R}$ 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, the algebraic structure of $\pi_1\mathcal{O}(f)$ was computed 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 $\pi_0\mathcal{S}(f,\partial\mathbb{M})$, 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 $\pi_1\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein bottle and projective plane.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
  Prawa autorskie (c) 2025 Iryna Kuznietsova, Sergiy Maksymenko
• Date:
  2025.02.13
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 65, No 1 (March 2025); 35 - 77;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/58805/41533 ;https://apcz.umk.pl/TMNA/article/view/58805/41534
• Subject:
  Diffeomorphism;
  Morse function;
  Milnor number;
  Möbius band;
  foliation;
  Primary: 57S05, 57R45;
  Secondary: 37C05
• Identifier: https://apcz.umk.pl/TMNA/article/view/58805
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other