On generalized configuration space and its homotopy groups

TL;DR

This paper introduces a generalized configuration space for subsets of vector or projective spaces, focusing on calculating their homotopy groups and exploring relationships with classical spaces and manifolds.

Contribution

It defines the generalized configuration space $W_{k,n}(M)$ and computes its fundamental and higher homotopy groups for specific cases involving spheres and projective spaces.

Findings

Fundamental groups of $W_{k,n}(\,\mathbb{R}P^m)$ are determined for special cases.

Homotopy groups of $W_{k,n}(S^m)$ and $W_{k,n}(\,\mathbb{R}P^m)$ are shown to be isomorphic.

Connections between homotopy groups of generalized configuration spaces and Stiefel manifolds are established.

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.