A note on fundamental dimensions of Whitney continua of graphs
type: Item
instance of: scholarly article
title: A note on fundamental dimensions of Whitney continua of graphs
zbMATH Open document ID: 0651.54002
author: Hisao Kato
published in: Journal of the Mathematical Society of Japan
publication date: 1989-01-01T00:00:00Z
review text: In a previous paper [Trans. Am. Math. Soc. 300, 367--381 (1987; Zbl 0621.54006)] the author defined an index \(n(G)\) for a graph \(G\) and we showed that if \(\omega\) is any Whitney map for \(C(G)\), then \(\text{Fd}\,\omega^{- 1}(t)\leq n(G)-1\) for each \(t\), where \(\text{Fd}\,\omega^{-1}(t)\) denotes the fundamental dimension of \(\omega^{-1}(t)\). In this paper we give a sharper result. We define a more precise index \(I(G)\) \((\leq n(G))\) for a graph \(G\) and show that \(\text{Fd}\,\omega^{-1}(t)\leq I(G)-1\) for any Whitney map \(\omega\) for \(C(G)\) and \(0\leq t\leq \omega (G)\). In general, \(I(G)\) is not equal to \(n(G)\).
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C56
zbMATH DE Number: 4062123
zbMATH Keywords: graph
zbMATH Keywords: fundamental dimension
zbMATH Keywords: index
zbMATH Keywords: Whitney map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04120243
OpenAlex ID: W2034523188
DOI: 10.2969/JMSJ/04120243
description: scientific article; zbMATH DE number 4062123