On the homotopy invariance of configuration spaces

For a closed PL manifold M , we consider the configuration space F (M, k) of ordered k -tuples of distinct points in M . We show that a suitable iterated suspension of F (M, k) 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 Poincar embedding theory and fiberwise algebraic topology.