Duality theorems for regular homotopy of finite directed graphs
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.
scientific article; zbMATH DE number 3845632