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On the topological computation of K_4 of the Gaussian and Eisenstein integers.

Gangl, Herbert and Dutour Sikiriˇc, M and Gunnells, P and Hanke, J and Schuermann, A and Yasaki, D (2019) 'On the topological computation of K_4 of the Gaussian and Eisenstein integers.', Journal of homotopy and related structures., 14 . pp. 281-291.


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:=−1−−−√ and ρ:=(1+−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.

Item Type:Article
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Publisher statement:This is a post-peer-review, pre-copyedit version of an article published in Journal of homotopy and related structures. The final authenticated version is available online at:
Date accepted:25 July 2018
Date deposited:19 December 2019
Date of first online publication:18 August 2018
Date first made open access:19 December 2019

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