A Note On Non-Isolated Real Singularities and Links

Abstract

For analytic map germs $f:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$ having an isolated critical value in the origin with $\dim V(f)>0$ and satisfying the transversality property of D.B. Massey we show that for $c>0$ a large enough constant, and $k$ a large enough natural number the local Milnor-Lê fibers of $f$ and of the isolated singularity $g=f-c({\sum }_{i=1}^n{x}_i^2{)}^k$ satisfies the following. There exists a homotopy equivalence between the negative Milnor-Lê fiber of $f$, to which a cobordism between its boundary and the link of $f$ has been adjoined, and the negative Milnor-Lê fiber of $g$.