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Can someone please clarify if there always exist regular neighborhoods for a properly embedded surface in a 3-manifold? More precisely, if $F$ is a properly embedded surface in a 3-manifold $M$ and I give a simplicial complex structure to $M$, then will $F$ automatically recieve a subcomplex structure (after probably finitely many barycentric subdivisions of the triangulation of $M$) ? If this is so, then we can obviously construct a regular neighborhood of $F$. But I am feeling unsure about it now due to the following example where $F$ might be too big to allow such a compatible subdivision of the triangulation of $M$:

Example: Consider a mobius band without a contractible neighborhood in a genus 1 handlebody. One can construct one as follows: First take a solid cylinder $D^2\times{[-1,1]}$ and look at the central strip. Now glue the ends of the cylinder with $180$ degrees twist to make it a genus 1 handlebody. This will glue the central strip with a twist to make it a mobius band and it has no small neighborhood in the ambient genus 1 handlebody. It seems to me that this mobius band will not have a subcomplex structure for any given simplicial complex structure on the genus 1 handlebody.

In case the answer to my question is yes, I would also like to ask if there are no issues regarding the orientability of $F$ and $M$ while considering regular neighborhoods. That is whether it matters for constructing such a neighborhood if $F$ is non-orientable but $M$ is orientable or vice-versa or other combinations.

I am unable to clarify this by looking at Hempel's book on 3-manifolds. It would be great if someone could elucidate this.

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The answer is yes, with the correct technical hypothesis of "local flatness". (Local flatness rules out, for example, the sort of behavior shown by the Alexander horned sphere.) You are correct to think that this is a foundational issue in three-manifolds. The reference you mention (Hempel) is also the correct one. You could perhaps look at Bing's book.

In your example the Mobius band does have a regular neighborhood. Note that the open regular neighborhood is homeomorphic to the normal bundle of the surface F in the three-manifold M. The regular neighborhood need not (and in your example, is not) be homeomorphic to the product FxI.

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  • $\begingroup$ Thanks for the reference. I will try to get hold of Bing's book. Do you think orientability of $F$ or $M$ is of no consequence whatsoever in constructing a regular neighborhood? $\endgroup$
    – Maharana
    Commented Dec 31, 2009 at 6:18
  • $\begingroup$ By the way, it is clear from your argument that there is no problem in the smooth category. So clubbing it with Moise's theorem that upto 3-dimensions, PL=Top=Smooth, should provide us with a regular neighborhood. Do you think this right? $\endgroup$
    – Maharana
    Commented Dec 31, 2009 at 12:44
  • $\begingroup$ 1) Orientability is not an issue here. 2) I am not using smoothness. The normal bundle is homeomorphic to the open regular neighborhood, but that is a consequence of the machinary, not an input to it. I think it might help you to draw a Mobius band, draw the core curve, and draw a regular neighborhood of the core curve. $\endgroup$
    – Sam Nead
    Commented Dec 31, 2009 at 14:25

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