A-classification of map-germs via $_V$K-equivalence

TL;DR

This paper introduces $_V$K-equivalence, a simplified approach to classify map-germs up to A-equivalence by leveraging K-equivalence, demonstrated through a concise classification of certain codimension 2 maps.

Contribution

It establishes a new $_V$K-equivalence method that simplifies A-equivalence classification, connecting it to K-equivalence and demonstrating its effectiveness.

Findings

$_V$K-equivalence simplifies A-equivalence classification.

The method provides shorter, more efficient proofs.

Successfully classifies certain codimension 2 maps.

Abstract

The classification of map-germs up to the natural right-left equivalence (also known as A-equivalence) is often complicated. Certainly it is more complicated than K-equivalence which is extremely easy to work with because the associated tangent spaces are not 'mixed' modules as they are in the A-equivalence case. In this paper we use a version of K-equivalence, denoted $_V$K-equivalence, that is defined using K-equivalences that preserve a variety in the source of maps to classify maps up to A-equivalence. This is possible through making clear the connection between the two equivalences - previous work by Damon mostly focussed on the relation between the codimensions associated to the maps. To demonstrate the power and efficiency of the method we give a classification of certain A$_e$-codimension 2 maps from $n$-space to $n+1$-space. The proof using $_V$K-equivalence is considerably shorter - by a wide margin - than one using A-equivalence directly.