Normalized Milnor Fibrations for Real Analytic Maps

Abstract

Milnor's fibration theorem and its generalizations play a central role in the study of singularities of complex and real analytic maps. In the complex analytic case, the Milnor fibration on the sphere is always given by the normalized map $f/|f|$. In contrast, for real analytic maps the existence of such a normalized Milnor fibration generally fails, even when a Milnor--Le fibration exists on a tube. For locally surjective real analytic maps $f:(R^n,0)->(R^k,0)$ with isolated critical value, the existence of a Milnor--Le fibration on a tube is guaranteed under a transversality condition. However, the associated fibration on the sphere need not be given by the normalized map $f/||f||$, unless an additional regularity condition (d-regularity) is imposed. In this paper we show that this apparent obstruction is not intrinsic. More precisely, we prove that for any such map satisfying the transversality property, there exists a homeomorphism $h:(R^k,0)->(R^k,0)$ of the target space such that the composition $h^{-1}f$ becomes d-regular. As a consequence, the normalized map $(h^{-1}f)/||h^{-1}f||$ defines a smooth locally trivial fibration on the sphere, which is equivalent to both the Milnor--Le fibration on the tube and the Milnor fibration on the sphere. Our result reveals a closer topological parallel between real and complex analytic singularities than previously recognized, without changing the topological type of the singularity.