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On the topological computation of \(K_4\) of the Gaussian and Eisenstein integers. (English) Zbl 1412.19003

The authors prove that the finite abelian groups \(K_4(\mathbb{Z}[\sqrt{-1}])\) and \(K_4(\mathbb{Z}[\frac{1+\sqrt{-3}}{2}])\) have no \(p\)-torsion for \(p\geq 5\). They do this by exploiting the known close relation of the \(K\)-theory groups to the homology of general linear groups over these rings and compute the relevant part of homology using Voronoi reduction theory for positive definite and Hermitian forms. As the authors point out, the vanishing of these two \(K\)-groups is now known thanks to M. Kolster’s results [Math. Ann. 323, No. 4, 667–692 (2002; Zbl 1007.11068)] expressing the orders of \(K_{4n}(R)\) in terms of values of \(L\)-functions when \(R\) is the ring of integers of a CM number field, as well as the more recent Rost-Voevodsky theorem establishing the Quillen-Lichtenbaum conjecture. However, the proof in the article proceeds directly from the description of the \(K\)-groups as homotopy groups of an infinite loop space.

MSC:

19D50 Computations of higher \(K\)-theory of rings
11F75 Cohomology of arithmetic groups

Citations:

Zbl 1007.11068

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References:

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