Note on simple stable maps of 3-manifolds into surfaces (Q1976576)

Many results in global singularity theory deal with the number of certain singular points of a smooth map between smooth manifolds. Particularly difficult cases are the ones on the number of so-called multi-singularities. Hardly any of these multi-singularity formulas are known when the dimension of the target space is lower than that of the source space. In the paper under review such a formula is proved for some special maps of 3-dimensional manifolds into surfaces. The maps in question are the so-called `simple stable maps' , i.e. those having singularities not worse than definite and indefinite folds. The singular points of such a map form a link in the source. The components of this link are classified as definite and indefinite folds, the latter further splits into two cases I and II according to global behaviour. The main theorem of the paper counts the number of the crossings of this link when mapped to the surface. It is proved that this number has the same parity as the number of indefinite folds of type II. The main tools of the proof are the Stein factorization and a generalization of the rotation number for families of immersed circles in a surface. The result has already been proved by totally different techniques by \textit{D. Chess} in case the target manifold is the plane.