Study of phantom maps

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In this reserach we studied phantom maps out of a single or iterated loop space Ω^kX of a simply connected finite complex X and related topics. We attacked the following problems raised by McGibbon in his survey paper.(1) Let X be a simply connected finete complex which is not contractibel Is the universal phantom map out of the loop space Ω^κX essential?(2) Does there exist a finite complex X and an essential phantom map from ΩX to a target of finite trpe?(3) For a nilpotent group G we denot П_pG_(p) by G. Let {G_n} be an inverse sequence of finitely geneated nilpotent groups ans δ_* : lim__←^1G_n →lim__←^1G_n be the induced map between lim__←^1 sets. If lim__←^1G_n is nontrivial, does it follow that δ_*^-1(y) is an infinite set for each y in lim__←^1G_n?As of the problem 1 we solved this problem except the case κ = 1. As of the problem 2 we obtained a negative answer when X is rationally elliptic or spaces are localized at a prime. As of problem 3 we obtained an affirmative answer.