On the homotopy invariance of configuration spaces

Abstract

For a closed PL manifold $M$, we consider the configuration space $F\left(M,k\right)$ of ordered $k$–tuples of distinct points in $M$. We show that a suitable iterated suspension of $F\left(M,k\right)$ is a homotopy invariant of $M$. The number of suspensions we require depends on three parameters: the number of points $k$, the dimension of $M$ and the connectivity of $M$. Our proof uses a mixture of Poincaré embedding theory and fiberwise algebraic topology.