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. 



Does a regular neighborhood always exist for a properly embedded surface in a 3-manifold? 
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](http://en.wikipedia.org/wiki/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](http://books.google.co.uk/books?id=NBCQ0a_cdXcC&printsec=frontcover&dq=inauthor%3A%2522R.+H.+Bing%2522&client=firefox-a&cd=2#v=onepage&q=&f=false).


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.


