First Order Semi-Local Invariants of Stable Maps of 3-Manifolds into the Plane

In the late 1980s, Vassiliev introduced new graded numericalinvariants of knots, which are now called Vassiliev invariantsor finite-type invariants. Since he made this definition, manypeople have been trying to construct Vassiliev type invariantsfor various mapping spaces. In the early 1990s, Arnold and Goryunovintroduced the notion of first order (local) invariants of stablemaps. In this paper, we define and study first order semi-localinvariants of stable maps and those of stable fold maps of aclosed orientable 3-dimensional manifold into the plane. Weshow that there are essentially eight first order semi-localinvariants. For a stable map, one of them is a constant invariant,six of them count the number of singular fibers of a given typewhich appear discretely (there are exactly six types of suchsingular fibers), and the last one is the Euler characteristicof the Stein factorization of this stable map. Besides theseinvariants, for stable fold maps, the Bennequin invariant ofthe singular value set corresponding to definite fold pointsis also a first order semi-local invariant. Our study of unstablefold maps with codimension 1 provides invariants for the connectedcomponents of the set of all fold maps. 2000 Mathematics SubjectClassification 57R45 (primary), 32S20, 58K15 (secondary).