HOPF CONSTRUCTIONS AND HIGHER PROJECTIVE-PLANES FOR ITERATED LOOP-SPACES

We define a category, H-p(n) (for each n and p), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the mod p Dyer-Lashof operations for n-fold loop spaces. The category H-p(n) is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, P-p(n), from H-p(n) to the homotopy category of spaces and show P-p(n) to be left adjoint to the n-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.