Topology of manifolds and global theory of singularities (Theory of singularities of smooth mappings and around it)

"Theory of singularities of smooth mappings and around it". November 25～29, 2013. edited by Takashi Nishimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This is a survey article on topological aspects of the global theory of singularities of differentiable maps between manifolds and its applications. The central focus will be on the notion of the Stein factorization, which is the space of connected components of fibers of a given map. We will give some examples where the Stein factorization plays essential roles in proving important results. Some related open problems will also be presented