Hopf constructions and higher projective planes for iterated loop spaces

Abstract

We define a category, H p n \mathcal {H}_p^n (for each n n and p p ), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the mod p \bmod p Dyer-Lashof operations for n n -fold loop spaces. The category H p n \mathcal {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 P_p^n , from H p n \mathcal {H}_p^n to the homotopy category of spaces and show P p n P_p^n to be left adjoint to the n 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.