高斯整数和爱森斯坦整数\(K_4\)的拓扑计算

IF 0.7 4区 数学 Q2 MATHEMATICS
Mathieu Dutour Sikirić, Herbert Gangl, Paul E. Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki
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引用次数: 2

摘要

在本文中,我们使用拓扑工具研究了\(R=Z[i]\)和\(R=Z[\rho ]\)的代数k群\(K_4(R)\)的结构,其中\(i := \sqrt{-1}\)和\(\rho := (1+\sqrt{-3})/2\)。利用\(n\le 5\)的\(\mathrm {GL}_n(R)\)同调群与相关分类空间的同调群之间的紧密联系,利用Voronoi的正定二次型约简理论和厄米形式计算前者,得到\(\mathrm {GL}_n(R)\)作用于的一个非常大的有限胞复合体。我们的主要结果是\(K_{4} ({\mathbb {Z}}[i])\)和\(K_{4} ({\mathbb {Z}}[\rho ])\)对于\(p\ge 5\)没有p-扭转。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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 \(p\ge 5\).

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来源期刊
CiteScore
1.20
自引率
0.00%
发文量
21
审稿时长
>12 weeks
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.
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