Description

• Title:
  Lift factors for the Nielsen root theory on $n$-valued maps
• Creator:
  Brown, Robert F.;
  Gonçalves, Daciberg Lima
• Description:
  A root of an $n$-valued map $\varphi \colon X \to D_n(Y)$ at $a \in Y$ is a point $x \in X$ such that $a \in \varphi(x)$. We lift the map $\varphi$ to a split $n$-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of $\varphi$. We describe the relationship between the root classes at $a$ of the lift factors and those of $\varphi$. We define the Reidemeister root number $\RR (\varphi)$ in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number $NR(\varphi)$, the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the $\Phi$-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if $Y$ is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of $\varphi$. If $X$ and $Y$ are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of $\varphi$ that is invariant under $\Phi$-relation and this implies that if its index is non-zero, then the root class is essential.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2023 Robert F. Brown, Daciberg Lima Gonçalves;
  https://creativecommons.org/licenses/by-nd/4.0
• Date:
  2023.02.26
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 61, No 1 (March 2023); 269 - 289;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/42812/34966 ;https://apcz.umk.pl/TMNA/article/view/42812/34967
• Subject:
  55M20;
  $n$-valued map;
  Nielsen number;
  root;
  lift factor;
  Reidemeister number;
  root class;
  index;
  root-uniform
• Identifier: https://apcz.umk.pl/TMNA/article/view/42812
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other