Regular homotopy for immersions of graphs into surfaces (Q2821901)

Immersions of the circle into the plane were studied by \textit{H. Whitney} [Compos. Math. 4, 276--284 (1937; Zbl 0016.13804)]. He showed that two such immersions are regularly homotopic if and only if their winding numbers coincide. In the past decades many generalizations of Whitney's result were obtained.NEWLINENEWLINEThe present paper deals with immersions of graphs into compact surfaces (with or without boundary) and considers a generalization of Whitney's theorem in this context. A convenient form of winding number for these immersions is introduced, and a sufficient and necessary condition for the existence of a regular homotopy between two such immersions (in certain situations) is given in terms of this winding number.