Infinite loop spaces and Neisendorfer localization

There is a localization functor $L$ with the property that $L(X)$ is the $p$-completion of $X$ whenever $X$ is a finite dimensional complex. This same functor is shown to have the property that $L(E)$ is contractible whenever $E$ is a connected infinite loop space with a torsion fundamental group. One consequence of this is that many finite dimensional complexes $X$ are uniquely determined, up to $p$-completion, by the homotopy fiber of any map from $X$ into the classifying space $B E$.