Examples of finitely determined map-germs of corank 2 from $n$-space to $(n+1)$-space

TL;DR

This paper provides new examples of finitely determined map-germs of corank 2 from 3-space to 4-space, supporting the Mond conjecture by verifying it for these and higher-dimensional cases using novel augmentation techniques.

Contribution

It introduces a new augmentation method to generate finitely determined map-germs in higher dimensions and verifies the Mond conjecture for these examples.

Findings

Examples of finitely determined map-germs satisfying the Mond conjecture.

Introduction of a new augmentation technique for generating map-germs.

Verification of the conjecture in dimensions (4,5) and (5,6).

Abstract

We produce new examples supporting the Mond conjecture which can be stated as follows. The number of parameters needed for a miniversal unfolding of a finitely determined map-germ from $n$-space to $(n+1)$-space is less than (or equal to if the map-germ is weighted homogeneous) the rank of the $n$th homology group of the image of a stable perturbation of the map-germ. We give examples of finitely determined map-germs of corank 2 from 3-space to 4-space satisfying the conjecture. We introduce a new type of augmentations to generate series of finitely determined map-germs in dimensions $(n,n+1)$ from a given one in dimensions $(n-1,n)$. We present more examples in dimensions $(4,5)$ and $(5,6)$ based on our examples, and verify the conjecture for them.