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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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Homology of loop space

I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available here). On page 8, he states the following theorem. Let $X$ be a space such that $\pi_1(X)$ is abelian and acts tri …