On the homotopy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for some orbit spaces X

TL;DR

This paper investigates the homotopy type of the homotopy fiber of the inclusion of configuration spaces into product spaces for certain orbit spaces, providing explicit descriptions under specific conditions.

Contribution

It describes the homotopy type of the homotopy fiber of the inclusion map for configuration spaces of orbit spaces, extending known results to new classes of spaces.

Findings

Homotopy type of the homotopy fiber characterized for certain orbit spaces.

Explicit description of the long exact sequence in homotopy for specific cases.

Results apply to universal covers and free actions of Lie groups on spheres.

Abstract

Under certain conditions, we describe the homotopy type of the homo-topy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for the n-th configuration space F\_n(X) of a topological manifold X without boundary such that dim(X) $\ge$ 3. We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space S^k/G of a tame, free action of a Lie group G on the k-sphere S^k. If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map F\_n(S^k/G) $\rightarrow$ $\prod$\_1^n S^k/G.