Study of phantom maps

在这项研究中，我们研究了单连通有限复数X的单循环空间Ω^KX的模映射及相关问题。我们讨论了McGibbon在他的综述文章中提出的下列问题：(1)设X是非压缩的单连通有限复形，则环空间Ω^κX外的泛幻象映射是本质的吗？(2)是否存在有限复形X和从ΩX到有限目标的本质幻象映射？(3)对于幂零群G，我们用G表示П_PG_(P)。设{G_n}是有限生成的幂零群的逆序列，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δ_*^-1(Y)是Lim_←^1G_n中每个y的无限集吗？对于问题1，除了κ=1的情形外，我们解决了这个问题。对于问题2，当X是有理椭圆的或者空间局部化为素数时，我们得到了否定的答案。对于问题3，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.