Some Remarks on Configuration Spaces

This paper studies the homotopy type of the configuration spaces Fn(X) by introducing the idea of configuration spaces of maps. For every map f : X → Y , the configuration space Fn(f) is the space of configurations in X that have distinct images in Y . We show that the natural maps Fn(X) ← Fn(f) → Fn(Y ) are homotopy equivalences when f is a proper cell-like map between d-manifolds. We also show that the best approximation to X → Fn(X) by a homotopy invariant functor is given by the n-fold product map.