  We give a new and simple proof for the computation of the oriented and the
unoriented fold cobordism groups of Morse functions on surfaces. We also
compute similar cobordism groups of Morse functions based on simple stable maps
of 3-manifolds into the plane. Furthermore, we show that certain cohomology
classes associated with the universal complexes of singular fibers give
complete invariants for all these cobordism groups. We also discuss invariants
derived from hypercohomologies of the universal homology complexes of singular
fibers. Finally, as an application of the theory of universal complexes of
singular fibers, we show that for generic smooth map germs g: (R^3, 0) -->
(R^2, 0) with R^2 being oriented, the algebraic number of cusps appearing in a
stable perturbation of g is a local topological invariant of g.
