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?

Thanks a lot in advance!

  • $\begingroup$ 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. $\endgroup$ – Neal Apr 12 at 18:38
  • 2
    $\begingroup$ 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. $\endgroup$ – Misha Apr 12 at 18:46
  • 2
    $\begingroup$ 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. $\endgroup$ – Misha Apr 12 at 18:54
  • 1
    $\begingroup$ 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? $\endgroup$ – Dennis Apr 12 at 23:07
  • $\begingroup$ Yes, this all works for non-orientable manifolds. $\endgroup$ – Misha Apr 12 at 23:25

Your Answer

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

Browse other questions tagged or ask your own question.