TITLE: Approach topological manifolds with smooth manifolds
QUESTION [6 upvotes]: 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.

REPLY [8 votes]: 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.