I am wondering if there exists a closed topological manifold for which Homeo$(M)$ is not locally path-connected. If $M$ admits a smooth structure, then one can prove that Homeo$(M)$ is in fact locally path-connected, but what about non-smooth topological manifolds. For instance, is the homeomorphism group of $E_8$ locally path-connected?
2026-03-25 22:10:48.1774476648
Is Homeo(M) locally path-connected for a general topological manifold?
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