It is a deep fact of low-dimensional manifold topology that the notion of isomorphism coincides for the three categories Top, PL, and Diff. In other words, every topological manifold of dimension $n\leq3$ has a unique smooth structure up to diffeomorphism.
How does one prove this? I'm looking for high-level overviews of the big-picture strategy — or strategies, if there is more than one approach, which I suspect there must be. I'd like just enough detail to understand the primary difficulties and how were they solved. In other words, I'm looking for the key insights and an outline of the argument — the non-trivial parts where most people would get stuck. For the rest, feel free to point to references.
I'm interested both in contemporary approaches (how a textbook or expository note written this year would do it) and how the result came together historically. I vaguely recall the names of Munkres and Moise here, but I don't know exactly what they proved or how they did it.
