Simplicity of augmentations of codimension 1 germs and by Morse functions

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 ${\mathcal{A}}_e$-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 $\mathcal{A}$-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 ${\mathbb{C}}^4$.