engleski

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

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

Cohomology of arithmetic groups ; Voronoi reduction theory ; Linear groups over imaginary quadratic fields ; K-theory of number rings

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