Introduction

Let M and N be manifolds, M compact and f : M → N. Assume that f defines an extra structure on any neighborhood of a point p ∈ M. The point p is said to be regular with respect to the extra structure if there exists a neighborhood Up of p such that for any q ∈ U there exists also a neighborhood Uq so that the extra structure is equivalent on the two neighborhoods. The study of the structure on Up leads to a local problem. The study of the decomposition of M in maximal subsets with an homogeneous structure leads to a global problem. One way to achieve the global problem is to associate a graph to this decomposition.

This method has been applied to the study of the global classification of flows and maps. The approach in each case goes as follows: Once the local and multi-local behaviour of the critical set has been described, the relevant global topological information is codified in a graph, possibly with labels in either the vertices, the edges, or both. The typical questions then are:

1. Determine all the (labelled) abstract graphs that can be associated to some of the objects under study (Realization Problem).

2. […]