In this paper we classify the singular fibres of stable maps of closed (possibly non-orientable) 4-manifolds into 3-manifolds up to the 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.
Takahiro YAMAMOTO. "Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications." J. Math. Soc. Japan 58 (3) 721 - 742, July, 2006. https://doi.org/10.2969/jmsj/1156342035