Duality theorems for regular homotopy of finite directed graphs
type: Item
instance of: scholarly article
title: Duality theorems for regular homotopy of finite directed graphs
zbMATH Open document ID: 0533.05054
DOI: 10.1007/BF02851148
author: Marco Burzio
author: Davide Carlo Demaria
publication date: 1982-01-01T00:00:00Z
review text: Let S be a topological space and G a finite directed graph. A function f: \(S\to G\) is called 0-regular (resp. \(0^*\)-regular) iff for all vertices v,\(w\in G\) such that \(v\neq w\) and \(v\mapsto w\), it is \(f^{-1}(v)\cap \overline{f^{-1}(w)}=\emptyset\) (resp. \(\overline{f^{-1}(v)}\cap f^{-1}(w)=\emptyset)\). By definition a 0-homotopy (resp. \(0^*\)- homotopy) is a 0-regulr \((0^*\)-regular) map \(f:\quad S\times I\to G.\) The corresponding homotopy sets are denoted by Q(S,G) resp. \(Q^*(S,G)\). The authors prove that Q(S,G) and \(Q^*(S,G)\) are naturally equivalent provided that S is normal and countable paracompact. An example, (13.5), shows that the sets Q(S,G) and \(Q^*(S,G)\) are not equivalent if no assumptions on the topology of S are made. Relative versions of this theorem are also proved.
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 3845632
zbMATH Keywords: regular function
zbMATH Keywords: sets of homotopy classes
reviewed by: Werner End
MaRDI profile type: Publication
cites work: Q5615282
cites work: A normalization theorem for regular homotopy of finite directed graphs
cites work: Q4155491
cites work: Q3851288
cites work: Teoremi di normalizzazione per l'omotopia regolare dei grafi
cites work: Q4121193
cites work: Q5821080
cites work: Q5571563
full work available at URL: https://doi.org/10.1007/bf02851148
OpenAlex ID: W3022854163
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3845632