Graphs of stable maps from 3-manifolds to 3-space (Q1949872)

A weighted graph associated with a stable map from an orientable 3-manifold to \(\mathbb R^3\) is constructed by taking as many edges as the number of singularities of the stable map (singularities are closed orientable 2-manifolds) and as many vertices as the number of components in the complement of the set of singularities. Each edge has been assigned a weight equal to the genus of the corresponding singular surface and each vertex has also been assigned some weight. These graphs have been introduced recently by \textit{C. Mendes de Jesus} and the authors in [Proc. Steklov Inst. Math. 267, 205--216 (2009; Zbl 1207.58029)]. In this paper more properties of stable maps and the associated graphs are studied. A theorem of Eliashberg about stably parallelizable, closed, connected manifold is interpreted in terms of such graphs. By using certain surgeries between the stable maps, it is shown that any bipartite graph is the graph of a stable map from the 3-sphere with handles to \(\mathbb R^3\). A criterion for an arbitrary graph is formulated, to check whether it is the graph of some stable map from a prefixed 3-manifold to \(\mathbb R^3\). Also a sufficient condition, for a graph to be the graph of a stable map from a stably parallelizable 3-manifold to \(\mathbb R^3\), is obtained using Heegaard splittings. The article (which is part of the doctorate thesis of Raúl Oset Sinha) is self-explanatory and elegantly written. Many examples are given to make the concepts more clear. Sufficiently many references are also given, which are helpful to investigate these concepts further.