  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.
