The Mislin genus, phantom maps and classifying spaces

Abstract

For a finite type, nilpotent space X, we prove that the cardinality of the set Ph( X, Y), where Ph(−, −) denotes homotopy classes of phantom maps, depends only on the Mislin genus of X, at least if Y has countable higher homotopy groups. In the special case where X = BG, the classifying space of a 1-connected Lie group G, and Y is the iterated loop space of a 1-connected, finite CW-complex, we prove the stronger result that the isomorphism class of the group Ph( X, Y) depends only on the Mislin genus of X. The latter strengthening depends on two results of independent interest 1. (i) Under a fairly mild connectivity condition on X, the torsionization of X, that is the homotopy fiber of a rationalization map X → X (0), is a Mislin genus invariant; 2. (ii) the torsionization of BG, localized away from a prime p, is homotopy equivalent to the plus construction applied to a space of the form BΛ, where λ is a suitable locally finite, perfect (discrete) group.