# Definition of Regular Neighborhood for Curves in Sg

All:

I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:

We have a collection of curves $c_i$; i=1,2,..,n. embedded in $Sg$, the orientable genus-g surface, with 2 boundary components , satisfying these properties:

i) $c_i$ intersects $c_{i+1}$ transversely in a single point, and the algebraic intersection $c_i \cap c_{i+1}$ is +1

ii)$c_i \cap c_j$ is empty for |i-j|>1, and

iii) The homology classes of the $c_i$ are linearly-independent.

The claim is then made that if n is even, the regular 'hood (neighborhood) has genus

$\frac{n-1}{2}$ ,and two boundary components, while if n is odd, the 'hood has $\frac {n}{2}$ and one boundary component if n is even.

My understanding of regular neighborhoods is limited; I have had trouble finding a precise definition for them; there is one for simplicial complexes, and I have also seen , I think, descriptions of regular neighborhoods which see to come down to being tubular 'hoods, but neither of these seems to apply. The case of the algebraic intersection has to see with a choice of orientation for the (tangent spaces at intersection points of the )$c_i$'s , so that I do not see how this would help.

I think the author (D.Johnson) is choosing the $c_i's$ in a way that consecutive ones are "perpendicular", in that planes containing them would be perpendicular. I think also the $c_j's$ are supposed to be some variant of a symplectic basis (def. as being a basis {x_i,y_i} for $H_1(S_g)$ so that $x_i$ intersects $y_j$ exactly once when i=j, and the $x_i$ do not intersect the $y_j$ otherwise.

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If $X$ is a subset of a manifold $M$ a regular neighbourhood of $X$ in $M$ usually means a neighbourhood of $X$ in $M$ which is a mapping cylinder $f : Y \to X$, such that the embedding $C_f \to M$ embeds $Y$ as a smooth (or whatever category you like to work in) submanifold of $M$. The image of the embedding $C_f \to M$ is the regular neighbourhood of $X$ in $M$.

So if you have a collection of transverse curves in a surface, $M$ would be your surface and $X$ would be the union of the curves.

I'm not sure if there is a strong convention for "spine of a surface" in the literature but when people say such phrases I interpret this to mean either the 1-skeleton of a CW-decomposition of the surface, or some regular neighbourhood of such a 1-skeleton, or some combinatorial way of encoding such a 1-skeleton. Usually people mean one of these three things.

FYI, I'm going to vote to close your duplicate question on mathoverflow. Your question is more appropriate for this forum.

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Thanks for the answer; could you bound the conditions for Y as you described it?. Still, AFAIK, questions on MO come from research papers; this question comes from a paper by D.Johnson in which he shows that the Torelli group is fin.gen.(for g>2) by twists on bounding pairs , so I am confused as to what posts belong where. Do you have the additional comments made in MO? –  gary Jul 17 '11 at 2:34
Still, thanks for telling me in advance; I have had posts deleted and have been often down-voted without receiving explanations. Also, I'm a bit confused; you use $C_f$ for mapping cylinder, when I usually see that as the notation for the mapping cone. Did you mean mapping cone? If you had some recommended source , it would be great. –  gary Jul 17 '11 at 3:01
The general rule is if you're not sure where to post it, post here. MO is really only for material you're certain is of interest to research mathematicians. So for a terminology question to be suitable for MO it usually needs to be more than a "what is" type question -- like attempts to generalize, or something like that. I think most often this regular neighbourhood terminology appears in PL-manifold topology texts, since that's a setting where it's relatively easy to prove regular neighbourhoods exist. I mean mapping cylinder, sorry I forget the conventional terminology people use. –  Ryan Budney Jul 17 '11 at 14:47
Thanks, Ryan; could you please give the conditions on Y? –  gary Jul 18 '11 at 0:04
Just in case, the Y is the one in the first paragraph of your answer. –  gary Jul 18 '11 at 1:15