Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set $Um_{n}(R)$ of all unimodular row vectors of length $n$ over a ring $R$ with a global action. Its homotopy groups $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector $K$ -theory groups $K_{i+1}(Um_{n}(R)),i\geqslant 0$ of $Um_{n}(R)$ . It is known that the homotopy groups $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group $\text{GL}_{n}(R)$ viewed as a global action are the Volodin $K$ -theory groups $K_{i+1,n}(R)$ . The main result of Part I is an algebraic construction of the simply connected covering map $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ where $\mathit{EUm}_{n}(R)$ is the path connected component of the vector $(1,0,\ldots ,0)\in Um_{n}(R)$ . The result constructs the map as a specific quotient of the simply connected covering map $St_{n}(R)\rightarrow E_{n}(R)$ of the elementary global action $E_{n}(R)$ by the Steinberg global action $St_{n}(R)$ . As expected, $K_{2}(Um_{n}(R))$ is identified with $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$ . Part II of the paper provides an exact sequence relating stability for the Volodin $K$ -theory groups $K_{1,n}(R)$ and $K_{2,n}(R)$ to vector $K$ -theory groups.

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全球行动和矢量理论

像抽象群、陪集空间和 G 模这样的纯代数对象没有像分析和拓扑对象那样的空洞概念。然而，为代数对象配备全局作用会揭示其中的漏洞，并且由于全局作用的同伦理论，这些漏洞可以像在拓扑空间的同伦理论中一样被描述和量化。由于第一作者，本文的第一部分首先回顾了全局动作的概念，并详细描述了附加到一般线性、基本和 Steinberg 群的全局动作。考虑到这些例子，我们描述了任意全局动作的基本同伦理论，构建了它们的同伦群，并重新审视了它们的覆盖理论。然后我们装备套装 $Um_{n}(R)$ 所有长度的单模行向量 $n$ 在戒指上 $R$ 采取全球行动。它的同伦群 $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ 被命名为矢量 $K$ - 理论团体 $K_{i+1}(Um_{n}(R)),i\geqslant 0$ 的 $Um_{n}(R)$ . 已知同伦群 $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ 一般线性群的 $\text{GL}_{n}(R)$ Volodin 被视为一项全球行动 $K$ - 理论团体 $K_{i+1,n}(R)$ . 第一部分的主要结果是简单连通覆盖图的代数构造 $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ 在哪里 $\mathit{EUm}_{n}(R)$ 是向量的路径连通分量 $(1,0,\ldots ,0)\in Um_{n}(R)$ . 结果将地图构造为简单连接覆盖地图的特定商 $St_{n}(R)\rightarrow E_{n}(R)$ 基本的全球行动 $E_{n}(R)$ 通过斯坦伯格全球行动 $St_{n}(R)$ . 正如预期的那样， $K_{2}(Um_{n}(R))$ 被识别为 $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$ . 本文的第二部分提供了与 Volodin 稳定性相关的确切序列 $K$ - 理论团体 $K_{1,n}(R)$ 和 $K_{2,n}(R)$ 向量 $K$ - 理论团体。