ON WEAK HOMOTOPY-EQUIVALENCES BETWEEN MAPPING SPACES

Let S-+(n) denote the n-sphere with a disjoint basepoint; We give cond itions ensuring that a map h : X --> Y that induces bijections of homo topy classes of maps [S-+(n), X] congruent to [S-+(n), Y] for all n gr eater than or equal to 0 is a weak homotopy equivalence. For this to h old, it is sufficient that the fundamental groups of all path-connecte d components of X and Y be inverse limits of nilpotent groups. This co ndition 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 assumpt ion 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 eac h map f, the class of f-local spaces is precisely the class of spaces orthogonal to f and f boolean AND S-+(n) for n greater than or equal t o 1 in the based homotopy category. This has useful implications in th e theory of homotopical localization. (C) 1998 Elsevier Science Ltd. A ll rights reserved.