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A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.

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Homotopy groups of Diff(X) and Homeo(X)

No, the statement about the kernel and cokernel being finite is not true. For a closed $d$-manifold, $d \neq 4$, smoothing theory identifies the homotopy fibre of $$B\mathrm{Diff}(M) \longrightarrow B …
Oscar Randal-Williams's user avatar