$\mathrm{GE}_2$-rings and a graph of unimodular rows

TL;DR

This paper explores the topological properties of a graph constructed from unimodular rows over a ring, establishing connections between graph connectivity, algebraic properties of the ring, and the fundamental group of a related complex.

Contribution

It characterizes when the graph is path-connected and when the clique complex is simply connected in terms of the ring being a $ ext{GE}_2$-ring or universal for $ ext{GE}_2$, linking algebraic and topological properties.

Findings

$ ext{GE}_2$-rings correspond to path-connected $ ext{graph}$

The clique complex is simply connected iff the ring is universal for $ ext{GE}_2$

Fundamental group of the complex relates to $K_2(2,A)$ modulo symbols

Abstract

For a commutative ring $A$ we consider a related graph, $\Gamma(A)$, whose vertices are the unimodular rows of length $2$ up to multiplication by units. We prove that $\Gamma(A)$ is path-connected if and only if $A$ is a $\mathrm{GE}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\Gamma(A)$, we prove that $Y(A)$ is simply connected if and only if $A$ is universal for $\mathrm{GE}_2$. More precisely, our main theorem is that for any commutative ring $A$ the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.