Topology of real Milnor fibration for non-isolated singularities

Abstract

We consider a real analytic map $F=(f_1,...,f_k):({\mathbb{R}}^n,0)\to ({\mathbb{R}}^k,0)$, $2\le k\le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic $f_I=(f_{i_1},...,f_{i_l}):({\mathbb{R}}^n,0)\to ({\mathbb{R}}^l,0)$, induced from $F$ by projections where $1\le l\le n-2$ and $I=\{i_1,...,i_l\}$, also satisfies Milnor's conditions (a) and (b). We give several relations between the Euler characteristics of the Milnor fibre of $F$, the Milnor fibres of the maps $f_I$, the link of $F^{-1}(0)$ and the links of $f_I^{-1}(0)$.