The Topology of the Milnor Fibration

The fibration theorem for analytic maps near a critical point published by John Milnor in 1968 is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic “folklore theorems” which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity ( X , 0 ̲ ) $$(X, \underline {0})$$ does not depend on the choice of functions that define it. We glance at the use of polar varieties to studying the topology of singularities, which springs from ideas by René Thom. We give an elementary proof of a fundamental “attaching-handles” theorem, which is key for describing the topology of the Milnor fibers. This is also related to the so-called “carousel”, that allows a deeper understanding of the topology of plane curves and has several applications in various settings. Finally we speak about Lê’s conjecture concerning map-germs , and about the Lê-Ramanujam theorem, which still is open in dimension 2.