  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 F: 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.

Algebraic Topology;Geometric Topology