A signature invariant for stable maps of 3-manifolds into surfaces

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 paper, we give an intrinsic formula for the invariant in terms of a certain linking form associated with a stable map for the 3-manifold. As a corollary, we get a signature formula for 4-manifolds with boundary in terms of their singular fibers of stable maps.