In 1999, a friend of mine, Kazuhiro Sakuma, kindly asked me to give a series of lectures in the Kwansai Seminar on Differential Analysis, held at the Kinki University, Japan. At that time, I was studying the global topology of differ- entiable maps of 4-dimensional manifolds into lower dimensional manifolds. Sakuma and I had obtained a lot of interesting results concerning the relation- ship between the singularities of such maps and the differentiable structures of 4-dimensional manifolds; however, our results were not based on a system- atic theory and were not satisfactory in a certain sense. So I was trying to construct such a systematic theory when I was asked to give lectures.
I wondered what kind of objects can reflect the global properties of man- ifolds. “Singularity” of a differentiable map can be such an object, but it is local in nature. I already knew that the notion of the Stein factorization played an important role in the global study of such maps; for example, refer to the works of Burlet–de Rham [7] or Kushner–Levine–Porto [28, 30]. Stein factorization is constructed by considering the connected components of the fibers of a given map.