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

Abstract

In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite‐type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study first order semi‐local invariants of stable maps and those of stable fold maps of a closed orientable 3‐dimensional manifold into the plane. We show that there are essentially eight first order semi‐local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi‐local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps. 2000 Mathematics Subject Classification 57R45 (primary), 32S20, 58K15 (secondary).