Fibration structures and formulae for the Euler characteristics of Milnor fibers (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.In this article, we review some fibration structure theorems (also called Milnor s fibration theorems) recently proved in the real and complex cases, in the local and global settings. We give several Poincaré-Hopf type formulae which relate the Euler-Poincaré characteristics of these fibers (also called Milnor fibers) and indices (topological degrees) of appropriate vector fields defined on spheres of radii small or big enough. Some of them are new formulas