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


Author: José Seade
Journal: Bull. Amer. Math. Soc. 56 (2019), 281-348
MSC (2010): Primary 32SXX, 14BXX, 57M27, 57M50; Secondary 32QXX, 32JXX, 55S35, 57R20, 57R57, 57R77
DOI: https://doi.org/10.1090/bull/1654
Published electronically: November 8, 2018
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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 twofold. On the one hand, it should serve as an introduction to the topic for nonexperts, 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.


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Additional Information

José Seade
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuidad Universitaria, 04510 Coyoacan, Mexico
Email: jseade@im.unam.mx

DOI: https://doi.org/10.1090/bull/1654
Keywords: Milnor fibration, Milnor number, L\^e numbers, vanishing cycles, equisingularity, Lipschitz, Chern classes, indices of vector fields, 3-manifolds, mixed singularities, linear actions, polar actions, homotopy spheres
Received by editor(s): June 15, 2018
Published electronically: November 8, 2018
Additional Notes: The author’s research was partially supported by CONACYT 282937, PAPIIT-UNAM IN 110517, and CNRS-UMI 2001, Laboratoire Solomon Lefschetz-LaSoL, Cuernavaca, Mexico.
Dedicated: To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.
Article copyright: © Copyright 2018 American Mathematical Society