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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).

References

1. 1.

Arlettaz, D.: The Hurewicz homomorphism in algebraic $$K$$-theory. J. Pure Appl. Algebra 71365体育网站(1), 1–12 (1991)

2. 2.

Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactifications of Locally Symmetric Varieties, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2010) (with the collaboration of Peter Scholze)

3. 3.

Bass, H., Tate, J.: The Milnor ring of a global field. In: Algebraic $$K$$-Theory, II: “Classical” Algebraic $$K$$365体育网站-Theory and Connections with Arithmetic (Proceedings of the Conference, Seattle, Wash., Battelle Memorial Institute, 1972). Lecture Notes in Mathematics, vol. 342, pp. 349–446. Springer, Berlin (1973)

4. 4.

Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973) (Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault)

5. 5.

Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1994) (corrected reprint of the 1982 original)

6. 6.

Dutour Sikirić, M., Gangl, H., Gunnells, P.E., Hanke, J., Schürmann, A., Yasaki, D.: On the cohomology of linear groups over imaginary quadratic fields. J. Pure Appl. Algebra 220(7), 2564–2589 (2016)

7. 7.

Elbaz-Vincent, P., Gangl, H., Soulé, C.: Quelques calculs de la cohomologie de $${\rm GL}_N({\mathbb{Z}})$$ et de la $$K$$-théorie de $${\mathbb{Z}}$$. C. R. Math. Acad. Sci. Paris 335(4), 321–324 (2002)

8. 8.

Elbaz-Vincent, P., Gangl, H., Soulé, C.: Perfect forms, K-theory and the cohomology of modular groups. Adv. Math. 245365体育网站, 587–624 (2013)

9. 9.

Haesemeyer, C., Weibel, C.A.: The Norm Residue Theorem in Motivic Cohomology. Annals of Mathematics Studies. Princeton University Press, Princeton (to appear)365体育网站

10. 10.

Koecher, M.: Beiträge zu einer Reduktionstheorie in Positivitätsbereichen I. Math. Ann. 141365体育网站, 384–432 (1960)

11. 11.

Kolster, M.: Higher relative class number formulae. Math. Ann. 323365体育网站(4), 667–692 (2002)

12. 12.

Lee, R., Szczarba, R.H.: The group $$K_{3}({\bf Z})$$ is cyclic of order forty-eight. Ann. Math. (2) 104(1), 31–60 (1976)

13. 13.

Lee, R., Szczarba, R.H.: On the homology and cohomology of congruence subgroups. Invent. Math. 33(1), 15–53 (1976)

14. 14.

Lee, R., Szczarba, R.H.: On the torsion in $$K_{4}({\mathbb{Z}})$$ and $$K_{5}({\mathbb{Z}})$$. Duke Math. J. 45(1), 101–129 (1978)

15. 15.

Quillen, D.: Finite generation of the groups $$K_{i}$$ of rings of algebraic integers. In: Algebraic $$K$$-Theory, I: Higher $$K$$-Theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 179–198. Springer, Berlin (1973)

16. 16.

Quillen, D.: Higher algebraic $$K$$-theory I. In: Algebraic $$K$$-Theory, I: Higher $$K$$365体育网站-Theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)

17. 17.

Rognes, J.: $$K_4({\mathbf{Z}})$$ is the trivial group. Topology 39365体育网站(2), 267–281 (2000)

18. 18.

Soulé, C.: On the $$3$$-torsion in $$K_4({\mathbf{Z}})$$. Topology 39365体育网站(2), 259–265 (2000)

19. 19.

Soulé, C.: The cohomology of $${\rm SL}_{3}({\mathbf{Z}})$$. Topology 17(1), 1–22 (1978)

20. 20.

Staffeldt, R.E.: Reduction theory and $$K_{3}$$ of the Gaussian integers. Duke Math. J. 46(4), 773–798 (1979)

21. 21.

Voevodsky, V.: On motivic cohomology with $${\bf Z}/l$$-coefficients. Ann. Math. (2) 174365体育网站(1), 401–438 (2011)

22. 22.

Voronoi, G.: Nouvelles applications des paramètres continues à la théorie des formes quadratiques 1: Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math. 133(1), 97–178 (1908)

23. 23.

Weibel, C.: Algebraic $$K$$-Theory of Rings of Integers in Local and Global Fields. Handbook of $$K$$365体育网站-Theory, vols. 1, 2, pp. 139–190. Springer, Berlin (2005)

24. 24.

365体育网站Yasaki, D.: Voronoi tessellation data. . Accessed 2018

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.

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.

Rights and permissions

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