我们将单连通有限CW复形的p-局部空间的全体粗略地分为例外复形和generic复形两部分。对generic复形我们将主要通过有理同伦论中的信息来试图理解所考虑空间的p-torsion信息，包括渐近增长性。对例外复形，有理同伦论中的信息可能不再具有决定性作用。特别是对有理可缩的空间，我们将探索所考虑空间同伦群的p-torsion信息，包括渐近增长性。特别我们将对特殊的例外复形，比如（广义）张复形，研究它们的多重smash积及其在渐近增长性研究中的应用。

In our point of view,p-localizations of simply connected finite CW complexes are divided as generic or exceptional cases where for a generic p, rational homotopy information implies p-torsion homotopy information while for exceptional p,relation between rational homotopy information and p-torsion homotopy information is more complicated. In this project,we try to obtain p-torsion homotopy information like asymptotics of p-torsion part of homotopy groups from its rational homotopy for generic p. For exceptional local space,like (generalized Chang complexes) we will study its iterated smash product and its application to the product decomposition of loop space and asymptotics of p-torsion part of homotopy groups,etc.