On the homotopy fibre of the inclusion map $$F_n(X)\hookrightarrow \prod _1^nX$$ F n ( X ) ↪ ∏ 1 n X for some orbit spaces X

Abstract

Under certain conditions, we describe the homotopy type of the homotopy fibre of the inclusion map \(F_n(X)\hookrightarrow \prod _1^nX\) for the nth configuration space \(F_n(X)\) of a topological manifold X without boundary such that \(\mathrm{{dim}}(X)\ge 3\). We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space \(\mathbb {S}^k/G\) 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 \(F_n(\mathbb {S}^k/G)\hookrightarrow \prod _1^n\mathbb {S}^k/G\).