On Milnor's fibration theorem and its offspring after 50 years

TL;DR

This paper reviews Milnor's fibration theorem, its historical development over 50 years, and explores current research, connections, and its significance in understanding the geometry and topology of analytic maps near critical points.

Contribution

It provides a comprehensive overview of Milnor's fibration theorem, including classical results, recent developments, and connections to other mathematical areas, serving as an accessible introduction and a survey of ongoing research.

Findings

Revisits classical theory in real and complex settings

Highlights recent research directions and connections

Provides extensive references for further study

Abstract

Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading.