Simplicity of augmentations of codimension 1 germs and by Morse functions

TL;DR

This paper characterizes when augmented map-germs, especially those from codimension 1 germs or Morse functions, are simple, providing classifications and explicit lists of such simple augmentations.

Contribution

It offers a complete characterization of simplicity for augmentations from codimension 1 germs or Morse functions, extending classifications of simple monogerms.

Findings

Complete characterization of simple augmentations from codimension 1 germs.

Explicit list of simple augmentations from a0a0a0 to a0a0a0.

Identification of all simple augmentations except one in Mond's classification.

Abstract

We study the simplicity of map-germs obtained by the operation of augmentation and describe how to obtain their versal unfoldings. When the augmentation comes from an $\mathscr{A}_e$-codimension 1 germ or the augmenting function is a Morse function, we give a complete characterisation for simplicity. These characterisations yield all the simple augmentations in all explicitly obtained classifications of $\mathscr{A}$-simple monogerms except for one ($F_4$ in Mond's list from $\mathbb{C}^2$ to $\mathbb{C}^3$). Moreover, using our results we produce a list of simple augmentations from $\mathbb{C}^4$ to $\mathbb{C}^4$.