Topology of real Milnor fibration for non-isolated singularities

Abstract

We consider a real analytic map \(F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)\), \(2 \le k \le n-1\), that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic \(f_I=(f_{i_1},...,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)\), 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 \(f_I^{-1}(0)\).