Good real pictures of corank one map germs from the $n$-space to the $(n+1)$-space

TL;DR

This paper investigates when real corank one map germs from n-space to (n+1)-space accurately reflect the topology of their complex counterparts, providing practical criteria and examples for such 'good' real pictures.

Contribution

The authors introduce a new practical sufficient condition for real germs to produce accurate topological images of complex germs, with illustrative examples.

Findings

New practical criterion for good real pictures of complex germs

Examples demonstrating the application of the criterion

Insights into the topology of corank one map germs

Abstract

We study corank one $A$-finite germs $f:(\mathbb{R}^n,0)\rightarrow (\mathbb{R}^{n+1},0)$ and their complexifications. More precisely, we study when these germs provide good real pictures of the complex germs, i.e., when there is a real deformation that has the same homology in the image (hence, homotopy) than the generic complex deformation. We give a new sufficient condition that can be computed in practice, as well as examples.