Milnor’s Fibration Theorem for Real and Complex Singularities

Milnor’s fibration theorem is a landmark in singularity theory; it allowed to deepen the study of the geometry and topology of analytic maps near their critical points. In this chapter we revisit the classical theory and we glance at some areas of current research. We start with a glimpse at the origin of the fibration theorem, which is motivated by the study of exotic spheres. We then discuss an elementary example where all the ingredients of the fibration theorem are described in simple terms, and we use this as a guideline all along the chapter. The first part concerns complex singularities, which is a fairly mature area of mathematics; we survey some of the main steps in this line of research and indicate a wide bibliography as well as relations with other chapters in this book. The second part concerns real singularities, a theory that still is in its youth, though it springs also from Milnor’s seminal work.