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On the topological computation of \(K_4\) of the Gaussian and Eisenstein integers

Abstract

In this paper we use topological tools to investigate the structure of the algebraic K-groups \(K_4(R)\) for \(R=Z[i]\) and \(R=Z[\rho ]\) where \(i := \sqrt{-1}\) and \(\rho := (1+\sqrt{-3})/2\). We exploit the close connection between homology groups of \(\mathrm {GL}_n(R)\) for \(n\le 5\) 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 \(\mathrm {GL}_n(R)\) acts. Our main result is that \(K_{4} ({\mathbb {Z}}[i])\) and \(K_{4} ({\mathbb {Z}}[\rho ])\) have no p-torsion for \(p\ge 5\).

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Notes

  1. 1.

    More precisely [5, VII.7] constructs a spectral sequence converging to the equivariant homology \(H^G_*(X, M)\) of a G-complex X with coefficients in a G-module M; the \(E^1\) page has a form similar to (4). One can formulate an analogous spectral sequence for the equivariant homology of a pair (XY) of G-complexes with \(E^1\) page (4365体育网站), cf. the remarks in [5, VII.7] in the paragraphs preceding equation (7.1).

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Acknowledgements

We thank Ph. Elbaz-Vincent for very helpful discussions. We also thank an anonymous referee for suggesting numerous improvements and corrections to our paper. This research was conducted as part of a “SQuaRE” (Structured Quartet Research Ensemble) at the American Institute of Mathematics in Palo Alto, California in September 2013. It is a pleasure to thank AIM and its staff for their support, without which our collaboration would not have been possible.

Author information

Correspondence to Herbert Gangl.

Additional information

MDS was partially supported by the Humboldt Foundation. PG was partially supported by the NSF under contract DMS 1101640 and DMS 1501832. The authors thank the American Institute of Mathematics where this research was initiated.

Communicated by Chuck Weibel.

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Dutour Sikirić, M., Gangl, H., Gunnells, P.E. et al. On the topological computation of \(K_4\) of the Gaussian and Eisenstein integers. J. Homotopy Relat. Struct. 14, 365体育网站281–291 (2019). http://doi.org/10.1007/s40062-018-0212-8

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Keywords

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

Mathematics Subject Classification

  • Primary 19D50
  • Secondary 11F75