Universal homotopy associative, homotopy commutative H-spaces and the EHP spectral sequence

Assume that all spaces and maps are localised at a fixed prime p. We study the possibility of generating a universal space U(X) from a space X which is universal in the category of homotopy associative, homotopy commutative H -spaces in the sense that any map f: X → Y to a homotopy associative, homotopy commutative H -space extends to a uniquely determined H-map f̄: U(X) → Y. Developing a method for recognising certain universal spaces, we show the existence of the universal space F 2(n) of a certain three-cell complex L. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the d 1 -differential of the EHP-spectral sequence valid in a certain range. © 2006 Cambridge Philosophical Society.