Toward construction of the universal complex of the solutions of differential equations

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1st order differential equations are considered as hypersurfaces in the 1st jet space and solutions of the 1st order differential equations are curves on the hypersurfaces. Since, the classification of divergence diagrams induces the classifications of a certaintype 1st order differential equations, we study smooth maps between surfaces. It is well known that a smooth map between surface is approximated by a smooth map, which is called a generic map, whose singular values set is a regular curve with some cusps. For a smooth map, generic maps which are homotopic to the smooth map and whose singular values setare the simplest, in some sense, are studied. Furthermore, for a smooth map between surface, the pairs of the numbers of singularities, cups and nodes, on the singular values set of generic maps homotopic to the smooth map are determined.