Topology of Real Milnor Fibrations for Non-Isolated Singularities

Abstract

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