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 \(p\ge 5\).

中文翻译：

---

关于高斯整数和爱森斯坦整数的$$ K_4 $$ K 4的拓扑计算

在本文中，我们使用拓扑工具来调查代数的结构ķ -基团\（K_4（R）\）为\（R = Z [I] \）和\（R = Z [\ RHO] \） ，其中\ （i：= \ sqrt {-1} \）和\（\ rho：=（1+ \ sqrt {-3}）/ 2 \）。我们利用\（\ mathrm {GL} _n（R）\）对\（n \ le 5 \）的同源性群与相关分类空间的相似性群之间的紧密联系，然后使用正定二次方的Voronoi约简理论来计算前者和Hermitian形式产生一个非常大的有限元复合体，\（\ mathrm {GL} _n（R）\）作用于此。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。