On weak homotopy equivalences between mapping spaces

Let S+n denote the n-sphere with a disjoint basepoint. We give conditions ensuring that a map h: X → Y that induces bijections of homotopy classes of maps [S+n, X] ≅ [S+n, Y] for all n ⩾ 0 is a weak homotopy equivalence. For this to hold, it is sufficient that the fundamental groups of all path-connected components of X and Y be inverse limits of nilpotent groups. This condition is fulfilled by any map between based mapping spaces h: map∗(B, W) → map∗(A, V) if A and B are connected CW-complexes. The assumption that A and B be connected can be dropped if W = V and the map h is induced by a map A → B. From the latter fact we infer that, for each map ƒ, the class of ƒ-local spaces is precisely the class of spaces orthogonal to ƒ and ƒ λ S+n for n ⩾ 1 i homotopy category. This has useful implications in the theory of homotopical localization.