A note on fundamental dimensions of Whitney continua of graphs
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 \(n(G)\).
scientific article; zbMATH DE number 4062123