A Horrocks-type theorem for even orthogonal $K_2$

TL;DR

This paper extends the Horrocks theorem to unstable even-dimensional orthogonal Steinberg groups, advancing the understanding of K2-analogues of Serre's problem beyond the linear case.

Contribution

It proves a Horrocks-type theorem for unstable even-dimensional orthogonal Steinberg groups, a key step in the K2-analogue of Serre's problem.

Findings

Established the Horrocks theorem for these groups

Provided foundational results for K2-analogue of Serre's problem

Extended classical theorems to new algebraic structures

Abstract

We prove the Horrocks theorem for unstable even-dimensional orthogonal Steinberg groups. The Horrocks theorem for Steinberg groups is one of the principal ingredients needed for the proof of the $\mathrm{K}_2$-analogue of Serre's problem, whose positive solution is currently known only in the linear case.