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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.

Severin Schraven
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1 Answers1

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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.

Moishe Kohan
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