How do I construct pairs of pants and morse functions on them?

Question

I'm learning about morse theory and one of the pictures that keeps popping up is that of a pair of pants. Unfortunately, these pairs of pants are only a doodle for me; I have no idea how to model these rigorously. In addition, I want to be able to use the height functor to analyse these objects. Where can I learn about these constructions?

Answer 1

You can build a parametric model of a pair of pants by taking a torus and restricting to one half (letting the longitude vary from 0 to $\pi$, for example), and then restricting to points ... let me just write it out.

$$ X(u, v) = ( (2 + \cos v) \cos u , (2 + \sin v) \sin u , \sin v) $$

Now restrict to points $-\frac{\pi}{2} \le u \le \frac{\pi}{2}$, and then restrict to points whose first coordinate is no greater than, say, $2.5$. The resulting shape is a pair of pants, and the first coordinate is a morse function on it.