On the category of Euclidean configuration spaces and associated fibrations

TL;DR

This paper computes the Lusternik-Schnirelmann category of ordered Euclidean configuration spaces and provides bounds and exact values for the unordered spaces and associated fibrations, advancing topological understanding.

Contribution

It calculates the LS-category of ordered configuration spaces and determines bounds and exact values for unordered spaces and fibrations, especially when n is a power of 2.

Findings

Computed LS-category of F(R^n,k)

Provided bounds for B(R^n,k) and fibrations

Determined exact categories when n is a power of 2

Abstract

We calculate the Lusternik-Schnirelmann category of the k-th ordered configuration spaces F(R^n,k) of R^n and give bounds for the category of the corresponding unordered configuration spaces B(R^n,k) and the sectional category of the fibrations pi^n_k: F(R^n,k) --> B(R^n,k). We show that secat(pi^n_k) can be expressed in terms of subspace category. In many cases, eg, if n is a power of 2, we determine cat(B(R^n,k)) and secat(pi^n_k) precisely.