Some notes on spaces realized as classifying spaces

In this work, we are concerned with the realization of spaces up to rational homotopy as classifying spaces. In this paper, we first show that a class of rank-two rational spaces cannot be realized up to rational homotopy as the classifying space of any n-connected and π-finite space for $n\ge 1$. We also show that the Eilenberg-Mac Lane space $K(Q^r,n)$ $(r\ge 2,n\ge 2)$ can be realized up to rational homotopy as the classifying space of a simply connected and elliptic space X if and only if X has the rational homotopy type of ${\prod }_r{S}^{n-1}$ with n even.