A Note On Non-Isolated Real Singularities and Links

TL;DR

This paper investigates the topology of real singularities with non-isolated critical points, establishing homotopy equivalences between Milnor fibers of certain analytic map germs under specific conditions.

Contribution

It demonstrates a homotopy equivalence between Milnor fibers of a real analytic map germ and a modified singularity, extending understanding of their topological structure.

Findings

Homotopy equivalence between Milnor fibers of f and g.

Existence of a cobordism linking the boundary and the link of f.

Results apply to singularities satisfying Massey's transversality property.

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-L\^e 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-L\^e fiber of $f$, to which a cobordism between its boundary and the link of $f$ has been adjoined, and the negative Milnor-L\^e fiber of $g$.