Study of phantom maps

本文研究了单连通有限复形X的单圈或迭代圈空间Ω^kX上的幻影映射及其相关问题。我们讨论了麦吉尔在他的调查报告中提出的下列问题。(1)设X是一个非收缩的单连通复形，那么圈空间Ω^κX外的泛幻影映射是必要的吗？(2)是否存在一个有限复形X和一个从ΩX到一个有限目标的本质幻影映射？(3)对于幂零群G，我们用G来表示n_pG_（p）.设{G_n}是n-生成幂零群的逆序列，δ_*：lim_<$^1G_n →lim_<$^1G_n是lim_<$^1集之间的诱导映射.如果lim__<$^1G_n是非平凡的，那么对于lim__<$^1G_n中的每一个y，δ_*^-1（y）是无限集吗？从问题1开始，我们解决了这个问题，除了κ = 1的情况。对于问题2，当X是有理椭圆的或空间局部化于素数时，我们得到了否定的答案。第三个问题，我们得到了肯定的答案。

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.