In order to study the double suspension  when localized at a prime ,Selick filtered  by some H-spaces which geometrically realize a natural Hopf algebra filtration of the cohomology . Later, Gray showed that the fiber  of  has an integral classifying space  and there is a homotopy fibration . In this talk I will introduce how to  correspondingly filter  in a manner which is compatible with Selick’s filtration and the homotopy fibration. Furthermore, I will talk about our results on the multiplicative properties and homotopy exponents of the spaces in the filtrations.