Abstract. 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. 1
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