From Euler class groups to Mennicke symbols and a monic inversion principle

TL;DR

This paper explores the relationship between Euler class groups and Mennicke symbols over regular domains, introduces a new map between them, and demonstrates the monic inversion principle's validity for Euler class groups, with applications to real varieties.

Contribution

It establishes a comparison map between Euler class groups and Mennicke symbols and proves the monic inversion principle for Euler class groups, advancing understanding in algebraic K-theory.

Findings

Defined a map from Euler class groups to Mennicke symbols in certain cases

Carried out computations on real varieties using the new map

Proved the monic inversion principle for Euler class groups

Abstract

Let $R$ be a regular domain of dimension $d\geq 2$ which is essentially of finite type over an infinite perfect field $k$. We compare the Euler class group $E^d(R)$ with the van der Kallen group $Um_{d+1}(R)/E_{d+1}(R)$. In the case $2R=R$, we define a map from $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$ and study it in intricate details. As application, this map enables us to carry out some interesting computations on real varieties, using some very basic arguments. The formalism required to carry out the above investigation also provides us a requisite tool to show that the monic inversion principle holds for the Euler class groups.