Configuration spaces on punctured manifolds (Q1859334)

Among others, the authors study homotopy properties of the fibration \[ {\mathbf F}_{k-r}(M_{r+1})\to{\mathbf F}_{k}(M_1)\to{\mathbf F}_r(M_1), \] where \(M\) is a (smooth) simply connected \(n+1 > 2\) manifold, and \({\mathbf F}_i(M_j)\) is the space of configurations of \(i\) points in \(M\) with \(j\) points removed. For example, in particular they find that a necessary condition for this fibration to be fiber homotopically trivial is that \(n=3\) or 7 and \(r=2\). Study of the Lie algebras, \(\pi_*({\mathbf F}_{k+1}(M_i))\), \(i=0,1\), is part of the investigation.