There exist topological manifolds which don't admit a smooth structure in dimensions > 3, but I haven't seen much discussion on homotopy type. It seems much more reasonable that we can find a smooth manifold (of the same dimension) homotopy equivalent to a given topological manifold. Is this true, or is there a counterexample?



Are topological manifolds homotopy equivalent to smooth manifolds?
It is false for compact manifolds in 4 dimensions. Freedman showed that there is a compact simply connected topological 4-manifold with intersection form E8, but Donaldson showed that there is no such smooth manifold. 


