Some remarks about $\rho$-regularity for real analytic maps

TL;DR

This paper explores the concept of $ ho$-regularity in real analytic map germs, its relation to Milnor fibrations, and provides criteria for their existence and limitations in different contexts.

Contribution

It offers a comprehensive overview of conditions like Thom regular stratification and Milnor condition (b) for analytic map germs, including strategies to verify these conditions.

Findings

Thom regular stratifications imply $ ho$-regularity and Milnor fibrations.

Milnor condition (b) is crucial for the existence of open book structures.

The paper discusses scenarios where these regularity conditions fail.

Abstract

In this paper, we discuss the concept of $\rho$-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function $\rho$, which guarantees $\rho$-regularity. Consequently, both conditions are crucial for the presence of open book structures and the Milnor fibrations. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.