Universal homotopy associative, homotopy commutative H-spaces
For any connected space X the James construction shows that ΩSX is universal in the category of homotopy associative H-spaces in the sense that any map f: X ® Y to a homotopy associative H-space factors through a uniquely determined H-map. Let p be a fixed prime number, and X a space localised at p. We study the possibility of generating a universal space U(X) from X which is universal in the category of homotopy associative, homotopy commutative H-spaces in the same way as the James construction of a connected space is universal in the category of homotopy associative H-spaces. We develop a method for constructing certain universal spaces. This method is used to show that the universal space U(X) exists for a certain three-cell complex X. We use this specific example to derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the d1-differential of the EHP-spectral sequence valid in a certain range. Finally, we apply the developed method to the family of certain two-cell complexes and obtain their universal spaces. This result generalises the result of Cohen, Moore, Neisendorfer and Gray on the universal space of an odd dimensional p-primary Moore space.