In Part I, a notion of a group operad is proposed and then a theory of group operads is developed, extending the classical theories of groups, spaces with actions of groups, covering spaces and classifying spaces of groups. Group operads apply to homotopy theory via their associated monads. As an application, all braid groups and symmetric groups are used to produce a free group model for the canonical stabilization of the double loop double suspension of a space. In Part II, a new idea is proposed to investigate operations on spaces admitting actions of an operad and to understand the global structures of homotopy groups. This new idea is established if the operad is equivalent to the classifying operad of a group operad, and thus in particular produces a conceptual description of the structures of the homotopy groups of double loop spaces.
operads, group operads, iterated loop spaces, homotopy groups, smash operations, Brunnian braids
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