  In this paper, we establish the rationality conjecture raised in \cite{FKS}
for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq
2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X,
\mathbb{Z})$ such that $u^k\not =0$. %which is nilpotent with order of
nilpotency equal to $k+1$. Next, we illustrate (topologically) our result by
giving the minimal cell structure of such a CW-complex whose cohomology is a
truncated polynomial algebra.
