# Does a regular neighborhood always exist for a properly embedded surface in a 3-manifold?

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.

• 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? – Maharana Dec 31 '09 at 6:18