Universal spaces of two-cell complexes and their exponent bounds

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.