On the Classification and Geometry of Corank 1 Map-Germs from Three-Space to Four-Space

Summary

Dedicated to Terry Wall on the occasion of his 60th birthday.

The classification of germs of maps plays an important role in singularity theory. Not only do we obtain specific examples to which existing theory may be applied, but new phenomena often emerge and motivate new ideas. Here we present a classification of the simple corank 1 map-germs (ℂ³, 0) → (ℂ⁴, 0) up to A-equivalence together with those corank 1 germs of A_e-codimension ≤ 4. The classification proceeds inductively at the jet-level using the technique of Complete Transversals developed by Bruce, Kirk and du Plessis [1], and the unipotent determinacy theorems of Bruce, du Plessis and Wall [2]. The calculations are somewhat intensive and are performed using the computer package Transversa l written by the second author [9, 10]. (This package can deal with numerous calculations central to classification and unfolding theory. Its main aim is to calculate tangent spaces to orbits in a given jet-space and implement the complete transversal classification technique. All the standard equivalence relations are covered but the main concern is for A classifications.)

We adopt the ideas developed by Mond for map-germs (ℂ², 0) 7rarr; (ℂ³, 0). (This classification began as a Ph.D. thesis supervised by Wall [16, 17] and has motivated a lot of work on the A-classification and geometry of map-germs (ℂⁿ, 0) → (ℂ^p, 0) for n < p.)