Infinite loop spaces with trivial Dyer-Lashof operations

Abstract

AbstractLetpbe any prime. It is well known that the modp Dyer-Lashof algebra acts trivially on the modphomology of an Eilenberg-MacLane space. The main result of this paper is a converse of this fact. Specifically, it is shown that any connected infinite loop space with trivial action of the modpDyer-Lashof algebra is (localized atp) homotopy equivalent to a product of Eilenberg-MacLane spaces. It is then shown that this equivalence does not necessarily respect the infinite loop structures involved.