  Let $X$ be a co-$H$-space of $(p-1)$-cell complex with all cells in even
dimensions. Then the loop space $\Omega X$ admits a retract $\bar A^{\min}(X)$
that is the evaluation of the functor $\bar A^{\min}$ on $X$. In this paper, we
determine the homology $H_*(\bar A^{\min}(X))$ and give the $\EHP$ sequence for
the spaces $\bar A^{\min}(X)$.
