UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS
Abstract
Let P 2 n +1 be a two-cell complex which is formed by attaching a (2 n + 1)-cell to a 2 m -sphere by a suspension map. We construct a universal space U for P 2 n +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 : P 2 n +1 → Y to a homotopy associative, homotopy commutative H -space Y extends to a uniquely determined H -map f¯ : 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 S 2 n +1 { p r } of the p r -power map on S 2 n +1 , and we reproduce Neisendorfer's result that S 2 n +1 { p r } is homotopy associative, homotopy commutative and that the p r -power map on S 2 n +1 { p r } is null homotopic.