Milnor fibrations and regularity conditions for real analytic mappings

TL;DR

This paper investigates conditions under which real analytic mappings with isolated critical values induce fibrations on spheres and tubes, extending previous results and clarifying the role of regularity conditions in such fibrations.

Contribution

It introduces a new regularity condition ensuring fibrations for real analytic maps with isolated critical values, extending and refining prior work.

Findings

The (m)-regularity condition guarantees fibrations on small spheres and tubes.

Fibrations induced by the map are equivalent under the regularity condition.

Results extend previous theorems to maps with isolated critical values.

Abstract

When f : R power n to R power p, is a surjective real analytic map with isolated critical value, we prove that the (m)-regularity condition (in a sense we define) ensures that f ||f|| is a fibration on small spheres, f induces a fibration on the tubes and both fibrations are equivalent. In particular, we make the statement of [12] more precise in the case of isolated critical point and we extend it to the case of an isolated critical value.