On the L\^e-Milnor fibration for real analytic maps

TL;DR

This paper investigates the topology of real analytic map-germs with isolated critical values, establishing conditions under which they admit a Lê-Milnor fibration in the tube, by comparing the topology of the map with its projections.

Contribution

It provides necessary and sufficient conditions for real analytic maps to have a Lê-Milnor fibration, linking the topology of the map with that of its coordinate projections.

Findings

Conditions for Lê-Milnor fibration in the tube

Comparison of topology between the map and projections

Characterization of maps with isolated critical value

Abstract

In this paper, we study the topology of real analytic map-germs with isolated critical value $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We compare the topology of $f$ with the topology of the compositions $\pi_i^* \circ f$, where $\pi_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the projections $(t_1, \dots, t_n) \mapsto (t_1, \dots, t_{i-1}, t_{i+1}, \dots, t_n)$, for $i=1, \dots, n$. As a main result, we give necessary and sufficient conditions for $f$ to have a L\^e-Milnor fibration in the tube.