On generalized configuration space and its homotopy groups

Abstract

Let $M$ be a subset of vector space or projective space. The authors define the \emph{generalized configuration space} of $M$ which is formed by $n$-tuples of elements of $M$ where any $k$ elements of each $n$-tuple are linearly independent. The \emph{generalized configuration space} gives a generalization of the classical configuration space defined by E.Fadell. Denote the \emph{generalized configuration space} of $M$ by $W_{k,n}(M)$. The authors are mainly interested in the calculation about the homotopy groups of generalized configuration space. This article gives the fundamental groups of generalized configuration spaces of $\mathbb{R}{P}^m$ for some special cases, and the connections between the homotopy groups of generalized configuration spaces of $S^m$ and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces $W_{k,n}(S^m)$ and $W_{k,n}(\mathbb{R}{P}^m)$ are isomorphic.

20 pages,2 figures

References in corpus (1)

• Non-Reidemeister Knot Theory and Its Applications in Dynamical Systems, Geometry, and Topology