Embedding a graph-like continuum in a surface
A topological space \(X\) is a graph-like space if there is a 0-dimensional subspace \(V\) (these are the vertices of \(X\)) of \(X\) so that \(X\setminus V\) is the disjoint union of connected open sets (these are the edges of \(X\)), each of which is a homeomorph of the real line. In a natural way graph-like continua generalize finite graphs to infinite, compact structures. In the present paper the authors investigate embeddings of graph-like continua in compact surfaces. Generalizing results for embeddings of finite graphs they establish the following facts: (i) every edge is in an open disc that meets the graph-like continuum precisely in that edge; (ii) there are natural analogues of face boundary walks; (iii) there is a graph-like continuum triangulating the same surface and containing as a sub-graph-like continuum the original embedded graph-like continuum; (iv) the face boundaries generate a subspace of the cycle space; (v) the quotient of the cycle space by the boundary cycles is the homology of the surface.
scientific article; zbMATH DE number 6699166
scientific article