Approach topological manifolds with smooth manifolds

Question

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological manifold with a smooth manifold such that they are homotopy equivalent? When M is compact, I think we can embed it into an Euclidean space then maybe we can isotopy it slightly so that it is deformed into a smooth manifold.

Answer 1

As surprising as it is, this is not true.

By the work of Freedman, a compact, simply-connected topological 4-manifold is determined up to homeomorphism by the intersection form on second cohomology together with a certain $\mathbb{Z}/2\mathbb{Z}$ invariant. Moreover, all unimodular intersection forms arise in this way.

By work of Donaldson, the intersection forms that arise from smooth manifolds are rather special. Since the intersection form is a homotopy invariant, it follows that there exist topological 4-manifolds that are not smoothable, even up to homotopy.

You can find more details at wikipedia article.