Global actions, K-theory and unimodular rows

TL;DR

This paper applies Bak's global action framework to unimodular rows, computing their topological invariants and linking these to stability questions in algebraic K-theory.

Contribution

It explicitly computes the fundamental group of the unimodular row global action and relates it to stability phenomena in K-theory.

Findings

Computed the path connected component of unimodular row global action.

Determined the fundamental group of the unimodular row global action.

Connected the fundamental group to stability questions in K-theory.

Abstract

Global actions were introduced by Bak in order to have a homotopy theory in a purely algebraic setting. In this paper we apply his techniques in a particular case: the (single domain) unimodular row global action. More precisely, we compute the the path connected component and fundamental group for the unimodular row global action. An explicit computation of the fundamental group of the (connected component of) unimodular row global action is closely related to stability questions in K-theory. This will be shown by constructing an exact sequence with the fundamental group functor as the middle term and having surjective stability for the functor K_2 on the left and injective stability for the functor K_1 on the right.