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The loop space $Ω_X$ of a pointed topological space $X$ is the space of based maps from the circle $\mathbb S^1$ to $X$ with the compact-open topology.

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Equivariant splitting of loop space of a suspension

It is well known, e.g. by Cohen's "A model for the free loop space of a suspension", that there is a stable splitting of the free loop space $\mathcal{L} \Sigma X $of the suspension $\Sigma X $ give …