TITLE: On weak homotopy equivalences between mapping spaces
AUTHORS: Carles Casacuberta (casac@mat.uab.es)
Jose L. Rodriguez (jlrodri@mat.uab.es)
ABSTRACT: Let $S^n_+$ denote the $n$-sphere with a disjoint
basepoint. We give conditions ensuring that a map $h: X\to Y$
that induces bijections of homotopy classes of maps
$[S^n_+,X]\cong [S^n_+,Y]$ for all $n\ge 0$ 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 $h: map_*(B,W)\to map_*(A,V)$ 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 $A\to B$.
From the latter fact we infer that, for each map $f$, the class of
$f$-local spaces is precisely the class of spaces orthogonal to $f$
and $f\wedge S^n_+$ for $n\ge 1$ in the based homotopy category.
This has useful implications in the theory of homotopical localization.
(This article will appear in Topology.)