On the toplogical computation of K4 of the Gaussian and Eisenstein integers

TL;DR

This paper employs topological methods and Voronoi's reduction theory to analyze algebraic K-groups of Gaussian and Eisenstein integers, establishing the absence of p-torsion for primes p >= 5.

Contribution

It introduces a novel topological approach to compute K_4 groups of specific algebraic integers and determines their torsion properties.

Findings

K_4(Z[i]) and K_4(Z[rho]) have no p-torsion for p >= 5

Used Voronoi's reduction theory to construct large cell complexes for computations

Connected homology groups of GL_n(R) with classifying spaces for K-theory

Abstract

In this paper we use topological tools to investigate the structure of the algebraic K-groups K_4 (Z[i]) and K_4 (Z[rho]), where i := sqrt{-1} and rho := (1+sqrt{-3})/2. We exploit the close connection between homology groups of GL_n(R) for n <= 5 and those of related classifying spaces, then compute the former using Voronoi's reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GL_n(R) acts. Our main result is that K_4 (Z[i]) and K_4 (Z[rho]) have no p-torsion for p >= 5.