On the homotopy fibre of the inclusion map $F_n(X) \longrightarrow \prod_1^n X$ for some orbit spaces $X$
Résumé
Under certain conditions, we describe the homotopy type of the homo-topy fibre of the inclusion map $F_n(X) \longrightarrow \prod_1^n X$ for the $n$th configuration space $F_n(X)$ of a topological manifold $X$ without boundary such that $\dim(X) ≥ 3$. We then apply our results to the cases where either the universal covering of $X$ is contractible or $X$ is an orbit space $\mathbb{S}^k/G$ of a tame, free action of a Lie group $G$ on the $k$-sphere $\mathbb{S}^k$. 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 $F_n(\mathbb{S}^k/G) \longrightarrow \prod_1^n \mathbb{S}^k/G$.
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