Fibration theorems à la Milnor for analytic maps with non-isolated singularities

Abstract

AbstractWe study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions $$(f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2$$ ( f , g ) : R n → R 2 of the type $$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$ ( f , g ) = ∑ i = 1 n a i x i p , ∑ i = 1 n b i x i q , where $$a_i, b_i \in {\mathbb {R}}$$ a i , b i ∈ R are constants in generic position and $$p,q \ge 2$$ p , q ≥ 2 are integers.