We study the topology of the fibers of real analytic maps ℝn → ℝp, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝn → ℝp, p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom af-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝn → ℝk with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers Ff and Ff,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map [Formula: see text] on Ff associated to g; as the number of critical points of [Formula: see text] on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and Ff,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.
On the Lê-Milnor fibration for real analytic maps
