Universal spaces of two-cell complexes and their exponent bounds

Let P2n+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 P2n+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: P2n+1 -> Y to a homotopy associative, homotopy commutative H-space Y extends to a uniquely determined H-map fmacr: U -> 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 S2n+1{p(r)} of the p(r)-power map on S2n+1, and we reproduce Neisendorfer's result that S2n+1{p(r)} is homotopy associative, homotopy commutative and that the p(r)-power map on S2n+1{p(r)} is null homotopic.