TITLE: Unique differentiable structure on homeomorphic manifolds in low dimension
QUESTION [7 upvotes]: When I took the introductory lectures to geometry and topology some years ago our professor mentioned the following result:

Let $n\in \{1, 2, 3 \}$ and $M, N$ be two $n$-dimensional differentiable manifolds. Then
  $$ M \text{ and } N \text{ homeomorphic} \Rightarrow  M \text{ and } N \text{ diffeomorphic.} $$

I could not find a reference, where this is acutally proved. I'd greatly appreciate if someone could provide my with a paper (or book) containing this result.
I read in the comment section of how to prove that every low-dimensional topological manifold has a unique smooth structure up to diffeomorphism? that this should be in Moise's book "Geometric Topology in Dimensions 2 and 3", however, I couldn't find such a statement in there.

REPLY [3 votes]: See Corollary on page 333 of J.Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Bull. Amer. Math. Soc.
Volume 65, Number 5 (1959), 332-334:

Every homeomorphism of smooth 3-manifolds can be approximated by diffeomorphisms. 

It is probably explained in more details in 
J.Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. (2) 72 (1960) 521--554.