Fibrations structure and degree formulae for Milnor fibers

TL;DR

This survey reviews recent theorems on fibrations (Milnor's fibrations) in real and complex cases, and presents Poincaré-Hopf type formulas linking the topology of Milnor fibers to vector field indices.

Contribution

It compiles recent results on fibrations and introduces new formulas relating Milnor fiber topology to vector field degrees.

Findings

Fibrations structures are established in various settings.

Poincaré-Hopf formulas connect Euler characteristic and vector field indices.

Results apply to local and global Milnor fibers in real and complex cases.

Abstract

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the Euler-Poincar\'e characteristic of these fibers (also called Milnor's fibers) and indices (topological degree) of appropriated vector fields defined on spheres of radii small or big enough.