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Yes, the manifold question is completely answered here. Namely $\exp_n X^k$ (where $X^k$ is a $k$-manifold) is a manifold if and only if $k=1, n=3$ or $k=n=2.$ This is Theorem 1.3 in the referenced paper. EDIT Also, of course if $n=1,$ though the authors overlook this...