Study on phantom maps

点击翻译按钮获取中文摘要

Through our research we obtained the following three major results.1. We proved that for every simply connected finite complex X there is a map which induces rational homotopy equivalence from the loop space on X to a product of some odd dimensional spheres and loop spaces on odd dimensional spheres. This result implies that the phantom map out of the loop space on a simply connected finite complex into a finite type space is trivial.2. If a space X satisfies one of the following three conditions, then the stable suspension order of the universal phantom map out of the space X is infinite.(1) The fundamental group of X is finite and homotopy groups are vanish in all sufficiently large dimensions.(2) X is the classifying space of non-trivial Lie group.(3) X is the loop space on the Lie group whose integral homology group has torsion element.3. The loop space on symplectic group Sp (n) is stably indecomposable if n is greater than or equal to 2.