Let [Formula: see text] be a subset of vector space or projective space. The authors define generalized configuration space of [Formula: see text] which is formed by [Formula: see text]-tuples of elements of [Formula: see text], where any [Formula: see text] elements of each [Formula: see text]-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of [Formula: see text] by [Formula: see text].For studying topological property of the generalized configuration spaces, the authors calculate homotopy groups for some special cases. This paper gives the fundamental groups of generalized configuration spaces of [Formula: see text] for some special cases, and the connections between the homotopy groups of generalized configuration spaces of [Formula: see text] and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces [Formula: see text] and [Formula: see text] are isomorphic.

中文翻译：

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关于广义配置空间及其同伦群

令 [公式：见正文] 为向量空间或投影空间的子集。作者定义了[公式：见文本]的广义配置空间，它由[公式：见文本]-[公式：见文本]元素的元组组成，其中每个[公式：见文本]的任何[公式：见文本]元素text]-tuple 是线性无关的。广义配置空间给出了 Fadell 经典配置空间和 Stiefel 流形的概括。用[公式：见文]表示[公式：见文]的广义配置空间。为了研究广义配置空间的拓扑性质，作者计算了一些特殊情况的同伦群。本文针对一些特殊情况给出了[公式：见正文]的广义配置空间的基本组，以及 [公式：见正文] 的广义配置空间的同伦群与 Stiefel 流形的同伦群之间的联系。还证明了广义配置空间[公式：见正文]和[公式：见正文]的较高同伦群是同构的。