On generalized configuration space and its homotopy groups

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.