Hopf constructions and higher projective planes for iterated loop spaces

Abstract

Abstract.We define a category, %fpn (for each n and p), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the mod/? Dyer-Lashof operations for «-fold loop spaces. The category &J/1 is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, P" , from ^p" to the homotopy category of spaces and show Pjj to be left adjoint to the rc-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.

Infinite loop spaces with trivial Dyer-Lashof operations

“…This section is basically a summary of some techniques developed in [11], which are the key tools used in the proof of the Main Theorem.…”

“…In order to understand the definition of an i/^-space and an H™-map that is used in this paper, it is necessary to know some facts about the May-Milgram model for Q"S n X (see [12,13]). These facts were also used to develop the main technical tools from [11] that are used in this paper. They are outlined in the remainder of this section.…”

“…They are outlined in the remainder of this section. The reader is urged to consult [11] for a more thorough description of what follows.…”

“…Loosely speaking, an i/^-map is a map which preserves H^ structures (up to homotopy), together with a choice of homotopy making the requisite diagram commute. For the precise definition of an //^-map, the reader is referred to [11] (see also [18] for a different, but essentially equivalent version when p = 2).…”

“…Let X be a path-connected //^-space. In [11], we defined a sequence of'projective spaces', PpX, for n S? 0.…”