MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

We study Milnor fibrations of real analytic maps (R(n), (0) under bar) ->(f) (R(p), 0), n >= p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f(-1) (0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom a(f)-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/parallel to f parallel to. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in R(n) the spherefication gives a fiber bundle equivalent to the Milnor fibration.