Hermitian K-theory and 2-regularity for totally real number fields

TL;DR

This paper determines the 2-primary torsion subgroups of hermitian K-groups for rings of 2-integers in totally real 2-regular number fields, revealing periodicity and confirming the hermitian Quillen-Lichtenbaum conjecture.

Contribution

It provides a complete description of 2-primary torsion in hermitian K-theory for specific number fields and establishes periodicity and conjecture validation.

Findings

2-primary torsion subgroups are fully determined

The result exhibits almost periodicity with period 8

The 2-primary hermitian Quillen-Lichtenbaum conjecture is proved

Abstract

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8. In both the orthogonal and symplectic cases, we prove the 2-primary hermitian Quillen-Lichtenbaum conjecture.