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.


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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.