Spatial-graph isotopy and the rearrangement theorem (Q1392658)

Let \(S(G)\) be the set of all (piecewise linear) embeddings \(\Gamma: G\to \mathbb{R}^3\) of a fixed graph \(G\) in \(\mathbb{R}^3\). According to \textit{K. Taniyama} [Topology 33, No. 3, 509-523 (1994; Zbl 0823.57006)], two elements \(\Gamma\), \(\Gamma' \in S(G)\) are said to be isotopic (resp. cobordant) if a level-preserving (resp. locally flat) PL-embedding \(\Phi:C \times I\to \mathbb{R}^3 \times I\) exists so that for some \(0<\varepsilon <{1\over 2}\), \(\Phi(x,t)= (\Gamma (x),t)\) if \((x,t)\in G\times [0,\varepsilon]\), \(\Phi(x,t)= (\Gamma' (x),t)\) if \((x,t)\in G \times [1- \varepsilon,1]\) and \(\Phi(G \times [\varepsilon,1-\varepsilon] \subset \mathbb{R}^3 \times [\varepsilon, 1- \varepsilon]\). The present paper is devoted to study the interactions between spatial graph isotopy and cobordism. In particular, a general ``Rearrangement Theorem on Spatial-Graph Isotopy'' is proved to hold: it enables to factorize every spatial graph isotopy by means of a finite sequence of moves of special type, called blowing down, followed by a finite sequence of moves of another special type, called blowing up. The usefulness of the above rearrangement was already pointed out in [\textit{T. Soma}, Topology Appl. 73, No. 1, 23-41 (1996; Zbl 0861.57005)] and in [\textit{Inaba} and \textit{T. Soma}, On spatial graph isotopic to planar embeddings, `Proceedings of Knots 96', World Scientific (1997)], where only partial similar results were obtained.