Homotopy and crossings of systems of curves on a surface
Let \(C_ 1,...,C_ k\) and \(C_ 1',...,C_ k'\) be primitive closed curves on a compact orientable surface S. In the paper, a necessary and sufficient condition in terms of the number of crossings is found for the existence of a permutation \(\pi\) of \(\{\) 1,2,...,k\(\}\) such that \(C'_{\pi (i)}\) is freely homotopic to \(C_ i\) or \(C_ i^{-1}\), \(i\in \{1,2,...,k\}\). The condition is shown to be equivalent to the non- singularity of a certain infinite symmetric matrix. An important role in the proofs is played by a graph embedded on S made up of the curves \(C_ i'\).
scientific article; zbMATH DE number 4101212