Good real images of complex maps

TL;DR

This paper investigates the homology and homotopy types of real map images and their complexifications, establishing conditions for when real perturbations mirror complex structures, with proofs of conjectures and applications.

Contribution

It proves a conjecture relating good real perturbations to complexification homotopy types and generalizes results across dimensions.

Findings

Good real perturbations can have the same homology as their complexification under certain conditions.

Proof of Marar and Mond's conjecture for singularities from c2b5^n to c2b5^{n+1}.

Applications to M-deformations and illustrative examples.

Abstract

We prove several results regarding the homology and homotopy type of images of real maps and their complexification. In particular, we study the local behavior of singular points after deformations. In this context, we prove a restrictive necessary condition for a real perturbation to have the same homology than its complexification, which is known as good real perturbation. We prove the conjecture of Marar and Mond stating that for singularities from $\mathbb{C}^n$ to $\mathbb{C}^{n+1}$, a good real perturbation is homotopy equivalent to its complexification, and show a generalization in other dimensions. Applications to $M$-deformations and other concepts as well as examples are given.