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

Gangl, Herbert; Dutour Sikiriˇc, M; Gunnells, P; Hanke, J; Schuermann, A; Yasaki, D

Abstract

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.

Citation

Gangl, H., Dutour Sikiriˇc, M., Gunnells, P., Hanke, J., Schuermann, A., & Yasaki, D. (2019). On the topological computation of K_4 of the Gaussian and Eisenstein integers. Journal of Homotopy and Related Structures, 14, 281-291. https://doi.org/10.1007/s40062-018-0212-8

• Journal Article Type: Article
• Acceptance Date: Jul 25, 2018
• Online Publication Date: Aug 18, 2018
• Publication Date: Mar 7, 2019
• Deposit Date: Aug 7, 2018
• Publicly Available Date: Dec 19, 2019
• Journal: Journal of homotopy and related structures.
• Print ISSN: 2193-8407
• Electronic ISSN: 1512-2891
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 14
• Pages: 281-291
• DOI: https://doi.org/10.1007/s40062-018-0212-8
• Public URL: https://durham-repository.worktribe.com/output/1324489
• Related Public URLs: https://arxiv.org/abs/1411.0584