  Let $P^{2n+1}$ be a two-cell complex which is formed by attaching a
$(2n+1)$--cell to a $2m$--sphere by a suspension map. We construct a universal
space $U$ for $P^{2n+1}$ in the category of homotopy associative, homotopy
commutative $H$--spaces. By universal we mean that $U$ is homotopy associative,
homotopy commutative, and has the property that any map $f\colon P^{2n+1}\lra
Y$ to a homotopy associative, homotopy commutative $H$--space $Y$ extends to a
uniquely determined $H$--map $\bar{f}\colon U\lra Y$. We then prove upper and
lower bounds of the $H$--homotopy exponent of $U$. In the case of a mod~$p^r$
Moore space $U$ is the homotopy fibre $S^{2n+1}\{p^r\}$ of the $p^r$--power map
on $S^{2n+1}$, and we reproduce Neisendorfer's result that $S^{2n+1}\{p^r\}$ is
homotopy associative, homotopy commutative and that the $p^r$--power map on
$S^{2n+1}\{p^r\}$ is null homotopic.
