# Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$

Hi 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 ci; i=1,2,..,n. embedded in Sg, the orientable genus-g surface, with 2 boundary components , satisfying these properties:

i) ci intersects ci+1 transversely in a single point, and the algebraic intersection ci∩ci+1 is +1

ii)ci∩cj is empty for |i-j|>1, and

iii) The homology classes of the ci 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.

To add some context, this paper is an effort to show that (here for the case of $S_{g,2}$ where g is the genus and 2 is the number of boundary components )the Torelli group is finitely-generated by "bounding pairs" (BP's) , meaning Dehn twists in opposite directions about the bounding curves.

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 )ci's , so that I do not see how this would help.

I think the author (D.Johnson) is choosing the ci′s in a way that consecutive ones are "perpendicular", in that planes containing them would be perpendicular. I think also the cj′s are supposed to be some variant of a symplectic basis (def. as being a basis {x_i,y_i} for H1(Sg) so that xi intersects yj exactly once when i=j, and the xi do not intersect the yj otherwise.

Edit: I hope this is not against MO protocol, but I thought I may tie-in another question: what is the spine of a surface? I think this is a generalization of the simplicial skeleton, but it would be nice to have something more precise.

I would interpret that as the union of regular neighbourhoods of the single curves: basically a union of bands $S^1\times (-1,1)$, one for each $c_i$, such that the intersection of any two is either empty or a square (i.e. the product of an interval in $S^1$ with $(-1,1)$, with the identification reversing the two factors). The genus count works well in this case. –  Marco Golla Jul 14 '11 at 23:22