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We review some of the problems where the graphs have been applied to the study of the global classification of stable maps.
AMS Classification: 57R45, 57M15, 57R65
Key words: Stable maps; foliations; graphs; global classification
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:
Determine all the (labelled) abstract graphs that can be associated to some of the objects under study (Realization Problem).
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