A Vassiliev type invariant of order one for stable maps of 3-manifolds into surfaces (Singularity theory of differential maps and its applications)

Abstract

Based on the signature formula for stable maps of closed oriented 4-manifolds into 3-manifolds, the author defined a Vassiliev type invariant of order one for stable maps of closed oriented 3-manifolds into surfaces. In this article, we announce an explicit formula for computing the invariant in terms of a certain linking matrix for the framed link consisting of regular fibers of a given stable map. As a corollary, we get a signature formula for 4-manifolds with boundary in terms of their singular fibers of certain "stable" maps. An enlightening explicit example is also given.