Milnor Fibrations and Regularity Conditions for Real Analytic Mappings

Khurram Shabbir\(^{a,*}\)

\(^a\)Department of Mathematics, Government College University, Katchery Road 54000, Lahore Pakistan.

Correspondence should be addressed to: dr.khurramshabbir@gcu.edu.pk

Copyright © 2023 Khurram Shabbir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 21 July 2023; Accepted: 01 September 2023; Published Online: 12 September 2023.

Abstract

When ƒ: \((\mathbb{R^n},0)\)→\((\mathbb{R^p},0)\)is a surjective real analytic map with isolated critical value, we prove that the(m)-regularity condition (in a sense we define) ensures that \(\frac{f}{||f||}\) is a fibration on small spheres, ƒ induces afibration on the tubes and these fibrations are equivalent.In particular, we make the statement of [12] more precise in the case of an isolated critical point and weextend it to the case of an isolated critical value.

Keywords:

Milnor Fibrations, Analytic Mapping, Regularity, Submersion