[eeereerews]

To emphasize where the topology comes in since it wasn't clear to me at first: the important bit is the saddles. Just before a saddle goes under, there are two important properties

1. there are land masses on both sides of the saddle. This would not be true if the polytope had holes.

2. if the lakes on either side of the saddle are joined up, the only way they can do that is if the lake circles around and encloses at least one of the two land masses (Jordan curve theorem). Once the saddle point goes under, that land mass is cut off from the "mainland". This would not be true if the polytope had a handle. The lakes could be joined up by going along the handle without enclosing one of the land masses.

https://i.imgur.com/I2FqwWQ.jpg