If $M$ is a non-compact smooth manifold, then $\text{Diff}(M)$ is not locally compact and hence not a manifold. Yet it still has manifold-like properties. In fact it has Lie group-like properties. It's "smooth" and a group. Yet it seems to be so big that not even a infinite-dimensional Lie group can capture its essence. What is it exactly?
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1Noncompactness of the manifold is irrelevant here. It's a long story. As for a partial answer with references see here. – Moishe Kohan Dec 10 '21 at 23:12