Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs (Q2492040)

One can consider maps between smooth manifolds that have some prescribed singularities, up to cobordism, where the singularities of the cobordisms are also prescribed. These cobordism theories are better understood if the target manifold has higher dimension than the source manifold. In the paper under review a new simple proof is given for the computation of the ``fold'' cobordism group of Morse functions on surfaces (both oriented and unoriented versions, together with the ``simple fold cobordism'' version). Moreover, the author studies the Vassiliev type universal complexes associated with the singular fibers of the maps considered. Using the cohomology of these complexes a complete set of cobordism invariants for the cobordism groups above are described. The hypercohomology version of Vassiliev's complex is also computed and the occuring invariants are compared to the invariants obtained by using only cohomology. Then the author applies the study of singular fibers to the stable perturbations of map germs. For example it is shown that for a generic smooth map germ from 3 dimensions to 2 dimensions (with the target oriented) the algebraic number of cusps in a stable perturbation is a local topological invariant of the germ. This invariant is strongly related to the cobordism invariants studied earlier.