# Existence of smooth structures on topological $3$-manifolds with boundary

It is said in this thread Unique smooth structure on $$3$$-manifolds that every topological $$3$$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have boundary or not and the references therein are hard to digest (since I am no expert in this field). So my question is whether every topological $$3$$-manifold with boundary admits a smooth structure?

By smooth structure I mean a collection of charts mapping homeomorphically onto open subsets of the (closed) upper half space, whose transition functions extend to smooth functions on some open neighbourhood of their domains.

I believe that the answer to my question is yes (assuming the theorem holds for topological $$3$$-manifolds without boundary) with the following reasoning:

Let $$M$$ be a topological $$3$$-manifold with non-empty boundary. Then we can consider the boundaryless double $$\widetilde{M}$$ of $$M$$, which is a topological $$3$$-manifold without boundary in which $$M$$ can be (topologically) embedded. We can now equip $$\widetilde{M}$$ with a smooth structure and restrict the charts to the embedding of $$M$$, which gives us a smooth atlas of the embedding and in turn a smooth atlas of $$M$$.

Is my reasoning correct?

• I think your argument has to be careful of counterexamples like the Alexander Horned Sphere, although I don't have time right now to precisely check if the AHS is indeed one. – Neal Apr 12 at 18:38
• Your argument is insufficient since it does not guarantee that $\partial M$ will be smooth in $\tilde{M}$. However, proofs of existence of a smooth structure work for manifolds with boundary as well. – Misha Apr 12 at 18:46
• Read projecteuclid.org/download/pdf_1/euclid.ijm/1256059559 and you will see that smoothing works even for manifolds with boundary. Munkres starts with the assumption that your manifold is triangulated. For triangulations of topological 3-manifolds (with boundary), read Moise's book. – Misha Apr 12 at 18:54
• Thank you for your answers, it already helped a lot! I have one last question: In the cited paper by Munkres it is stated at the beginnign of the first chapter that the manifold $M$ is assumed to be oriented. In remark 2.9 he then says that the results so far also hold for the non-orientable case. Does that mean that the upcoming theorem 2.10 also holds for non-orientable manifolds? – Dennis Apr 12 at 23:07
• Yes, this all works for non-orientable manifolds. – Misha Apr 12 at 23:25