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Group Operads and Homotopy Theory

We introduce the classical theory of the interplay between group theory and topology into the context of operads and explore some applications to homotopy theory. We first propose a notion of a group operad and then develop a theory of group operads, extending the classical theories of groups, spaces with actions of groups, covering spaces and classifying spaces of groups. In particular, the fundamental groups of a topological operad is naturally a group operad and its higher homotopy groups are naturally operads with actions of its fundamental groups operad, and a topological $K(π,1)$ operad is characterized by and can be reconstructed from its fundamental groups operad. Two most important examples of group operads are the symmetric groups operad and the braid groups operad which provide group models for $Ω^{\infty} Σ^{\infty} X$ (due to Barratt and Eccles) and $Ω^2 Σ^2 X$ (due to Fiedorowicz) respectively. We combine the two models together to produce a free group model for the canonical stabilization $Ω^2 Σ^2 X \hookrightarrow Ω^{\infty} Σ^{\infty} X$, in particular a free group model for its homotopy fibre.