Let M and M′ be unstable
modules over the modp Steenrod algebra such that there are spaces Y and
Y ′ with H∗(Y ;Zp) = U(M) and H∗(Y ′;Zp) = U(M′). Here U() is the
free-associative-graded-commutative-unstable algebra functor introduced by
Steenrod. Suppose g : M′→ M is a morphism of unstable modules. We
develop an obstruction theory which decides when g can be realized by a map
G : Y(p)→ Y(p)′, that is, g = H∗(G,Zp)|M′. We then apply this obstruction theory
to obtain p-equivalences of certain H-spaces with products of spheres and sphere
bundles over spheres which are determined by the cohomology structure of the
H-space.
