This is a very basic question:
I don't see the following:
If I cut a surface with boundary along non-contracible cycles into components with genus zero, how can those components have an unbounded number of boundary cycles?
Thank you
This is a very basic question:
I don't see the following:
If I cut a surface with boundary along non-contracible cycles into components with genus zero, how can those components have an unbounded number of boundary cycles?
Thank you
Okay, I see what you're referring to now. The authors are addressing the issue that if you only cut along non-contractible curves, you could be cutting off annuli. This happens if your curve is "parallel" to a boundary curve. So if you repeatedly cut along curves that are parrallel to the boundary, you'll create possibly an endless list of annuli without ever simplifying the original surface.
If $S_{g,b}$ is a connected surface of genus $g$ with $b$ boundary components, and you cut it along a curve $C$, there are two possibilities:
In particular, if you cut along a boundary-parallel curve you cut $S_{g,b}$ into an $S_{0,2}$ and an $S_{g,b}$.
The first case is when the curve $C$ is non-separating, the 2nd case is when $C$ is separating.