Some remarks on configuration spaces

Abstract

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

On Serre Microfibrations and a Lemma of M. Weiss

“…For finite-dimensional compact metric spaces, being cell-like is equivalent to having trivial shape [5]. We obtain the following result which generalises [8,Theorem 4.5].…”

“…REMARK 3.2. In the case where Y is also a topological d-manifold, Proposition 3.1 specialises to a different proof of [8,Theorem 4.5]. The statement in [8] only requires that f is proper and cell-like, but cell-like maps in this case are indeed cellular.…”

“…In the case where Y is also a topological d-manifold, Proposition 3.1 specialises to a different proof of [8,Theorem 4.5]. The statement in [8] only requires that f is proper and cell-like, but cell-like maps in this case are indeed cellular. This is a consequence of the cellularity criterion of McMillan [6] for dimension d = 4, and Repovš [9] for d = 4 (see also Lacher [4,Theorem 4.3]).…”

“…Relative configuration spaces. In [8], we considered configuration spaces associated with maps as an attempt to deal with the lack of functoriality. We recall that given a space X, the space of (ordered) configurations of n points in X is defined to be the subspace F n (X) ⊂ X n consisting of the n-tuples of pairwise distinct points.…”

“…The map f * : F n (f ) → F n (Y ) is again proper and cell-like/cellular, and therefore a (proper) homotopy equivalence, see [8,Proposition 4.4]. REMARK 3.2.…”