Description

• Title:
  On representation of the Reeb graph as a sub-complex of manifold
• Creator:
  Marzantowicz, Wacław;
  Silva, Nelson;
  Kaluba, Marek
• Description:
  The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$\nobreakdash-dimensional complex $\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f) $. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $\pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2015 Topological Methods in Nonlinear Analysis
• Date:
  2015.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 45, No 1 (March 2015); 287 - 308;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.015/5350
• Subject:
  Reeb graph;
  critical point;
  gradient flow;
  Primary 57N65;
  57R70;
  Secondary 57M50;
  58K65
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.015
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other