On the topological computation of $$K_4$$K4 of the Gaussian and Eisenstein integers

Abstract

In this paper we use topological tools to investigate the structure of the algebraic K-groups $$K_4(R)$$K4(R) for $$R=Z[i]$$R=Z[i] and $$R=Z[\\rho ]$$R=Z[ρ] where $$i := \\sqrt{-1}$$i:=-1 and $$\\rho := (1+\\sqrt{-3})/2$$ρ:=(1+-3)/2. We exploit the close connection between homology groups of $$\\mathrm {GL}_n(R)$$GLn(R) for $$n\\le 5$$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 $$\\mathrm {GL}_n(R)$$GLn(R) acts. Our main result is that $$K_{4} ({\\mathbb {Z}}[i])$$K4(Z[i]) and $$K_{4} ({\\mathbb {Z}}[\\rho ])$$K4(Z[ρ]) have no p-torsion for $$p\\ge 5$$p≥5.