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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

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


Authors

M Dutour Sikiriˇc

P Gunnells

J Hanke

A Schuermann

D Yasaki



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
Related Public URLs https://arxiv.org/abs/1411.0584

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