Mathieu Dutour Sikirić, Herbert Gangl, Paul E. Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki
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On the topological computation of \(K_4\) of the Gaussian and Eisenstein integers
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 \(p\ge 5\).
期刊介绍:
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.