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 $c_i$ for $i=1,2,..,n$ embedded in the orientable genus-$g$ surface with 2 boundary components $S_{g,2}$ satisfying:

  1. $c_i$ intersects $c_{i+1}$ transversely in a single point, and the algebraic intersection $c_i\cap c_{i+1}$ is $+1$
  2. $c_i\cap c_j$ is empty for $|i-j|>1$
  3. the homology classes of the $c_i$ are linearly-independent.

The claim is then made that the regular neighborhood has genus $\frac{n−1}{2}$, and two boundary components if $n$ is odd, and it has genus $\frac{n}{2}$ and one boundary component if $n$ is even.

Context: This concerns a paper where it is shown that the Torelli group for $S_{g,2}$ is finitely-generated by "bounding pairs," 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 neighborhoods. However, neither of these seems to apply. The algebraic intersection has to see with a choice of orientation for the (tangent spaces at intersection points of the) $c_i$'s , so 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", so that planes containing them would be perpendicular. I think also the $c_j$′s are supposed to be some variant of a symplectic basis (definition is being a basis $\{x_i,y_i\}$ for $H_1(S_{g,2})$ so that $x_i$ intersects $y_j$ exactly once when $i=j$, and the $x_i$ do not intersect the $y_j$ 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.

  • $\begingroup$ 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. $\endgroup$ – Marco Golla Jul 14 '11 at 23:22
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    $\begingroup$ I'm voting to close as this is a duplicate of your question on math.stackexchange. See: math.stackexchange.com/questions/51484/… $\endgroup$ – Ryan Budney Jul 15 '11 at 2:28

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