The Vaserstein symbol on real smooth affine threefolds

TL;DR

This paper characterizes when the Vaserstein symbol is injective on smooth affine real threefolds, showing it is bijective precisely when the real points have no compact connected components.

Contribution

It provides a topological criterion for the injectivity and bijectivity of the Vaserstein symbol on smooth affine real threefolds.

Findings

Vaserstein symbol is bijective iff real points have no compact connected components.

The paper establishes a necessary and sufficient topological condition.

Connects algebraic K-theory with real algebraic topology.

Abstract

We give a necessary and sufficient topological condition for the Vaserstein symbol to be injective on smooth affine real threefolds. More precisely, we show that the Vaserstein symbol is a bijection for such a threefold X if and only if the set of compact connected components of the real points of X (endowed with the Euclidean topology) is empty.