Phantom maps and injectivity of forgetful maps (Q5960532)

Let \(P\) be the total space of a principal \(G\)-bundle, and assume that \(P\) has the homotopy type of a simply connected finite CW complex. The associated forgetful map \(F\) is the natural homomorphism from the group of homotopy classes of equivariant self-equivalences of \(P\) to the group of homotopy classes of all self-equivalences. By constructing an exact sequence, the authors show that the kernel of \(F\) is the quotient of the fundamental group of a mapping space \(M\) by a countable subgroup. By using the theory of phantom maps they obtain conditions for \(\pi_1 M\) to be trivial (in which case \(F\) is injective) or uncountable (in which case \(F\) is not injective). The injectivity of forgetful maps is also related to the Halperin conjecture in rational homotopy theory.