Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications

Abstract

In this paper we classify the singular fibres of stable maps of closed (possibly non-orientable) 4-manifolds into 3-manifolds up to the $C^{\mathrm{\infty }}$ equivalence. Furthermore, we obtain several results on the co-existence of the singular fibres of such maps. As a consequence, we show that under certain conditions, the Euler number of the source 4-manifold has the same parity as the total number of certain singular fibres. This generalises Saeki's result in the orientable case.