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Do you know a good reference for the existence and uniqueness of a smooth structure on $3$-manifolds?

As far as I understand topological $3$-manifolds admit a unique smooth structure. I could find the following references for this result:

It follows from Hauptvermutung for $3$-manifolds (Theorems 3 and 4 in [2]) and from the fact that a combinatorial $3$-manifold has a unique smoothing (see Theorem 4.2 in [1]).

However, I am not quite satisfied with this answer since it requires a good understanding of what is written in these papers. For someone like me, who does not know geometric topology well, it would be better to have a reference to an explicit statement.

[1] M. W. Hirsch, B. Mazur, Smoothings of piecewise linear manifolds. Annals of Mathematics Studies, No. 80. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.

[2] E. E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. of Math. 56 (1952), 96-114.

There is a related post, but I do not find the answer posted there satisfactory. Smooth structures on closed $3$-manifolds are unique up to diffeomorphism? I think Moise does not talk about smooth structures only about triangulations.

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An alternative to Moise's paper for the existence and uniqueness of piecewise linear (PL) structures on topological 3-manifolds is the paper "The triangulation of 3-manifolds" by A.J.S. Hamilton in Quart. J. Math. Oxford (2), 27 (1976), 63-70. The result is stated as Theorem 2 there and proved in the rest of the paper using the famous Kirby torus trick together with several basic results about PL 3-manifolds.

For the existence and uniqueness of smooth structures on PL 3-manifolds there is a nice exposition in section 3.10 of Thurston's book "Three-Dimensional Geometry and Topology".

Added in 2024: Since writing the above (in 2018) I have looked again at Hamilton's paper and there is one point in the proof where I find that I am unable to fill in the omitted details. In fact, I am not sure that I fully understood this step when I first looked at the paper long ago. This was when I was presenting the proof in a class around 1983. I still have my class notes for the course but they seem to stumble at this point.

The point in question occurs in the middle of page 67 of Hamilton's paper where the author says "it is not difficult to see that ...". Hamilton wrote the paper as an undergraduate working with Peter Scott, but then he left mathematics so it is unlikely he would remember the argument 50 years later, even if he could be located. I asked Peter Scott about this in 2023 but he was too ill to provide any assistance and died not long after. If anyone reading this posting has read and understood this step of Hamilton's proof I would greatly appreciate hearing about it. (I should post this as a separate question.)

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    $\begingroup$ I suggest asking this as a separate question. Otherwise, I suspect that most people won't notice it. $\endgroup$
    – Andy Putman
    Commented Jul 11 at 19:12
  • $\begingroup$ Another source is Moises book on 3-manifolds. I think the whole point of the book was to give a clean proof of the triangulation theorem. $\endgroup$
    – Moishe Kohan
    Commented Jul 11 at 19:28
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    $\begingroup$ @MoisheKohan: Yes, but Hamilton's proof is much shorter and more efficient (at least as I understand it, which is only in outline). $\endgroup$
    – Andy Putman
    Commented Jul 11 at 19:34

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