If you want to get an idea of the famous PoincarĂ© conjecture for which Grigoriy Perelman was recently awarded a Millenium prize, I highly recommend watching this very entertaining lecture by Curtis McMullen at Harvard in 2006.

I had never realised that tori of genus greater than one are hyperbolic! It makes sense, if you think that a sphere (genus 0) has positive curvature, a torus (genus 1) is flat, that tori of genus 2 and more have negative curvature. These higher tori can be constructed by gluing together the edges of polygons.

I wonder whether anti-de Sitter space can also be viewed as a high genus space. Are the many black holes in the universe implying that it has a high "genus"? But the notion of genus isn't clear in higher dimensions...

So I started (very naively) to think about black holes on a latex surface, like the ones shown in the lecture. Consider a sphere and put a stone at the north pole. The sphere will bend, just like spacetime bends around the sun. Now imagine that you had a way to increase the mass of the stone, it would create a well at the north pole which would eventually touch the south pole from the inside. What happens then? The well could continue to become deeper and deeper but now it would create a spike out of the south pole. This is certainly misleading. More likely, once the stone approaches the south pole from the inside, its mass sucks it in, creating a well (as well).

Ultimately, what you'll end up with is a torus. The Planckian mass is at the centre of this torus, where there is really no spacetime: that's the black hole!

I had an unexpected confirmation of this picture in my kitchen. There was a frying pan full of a layer of greasy water and I noticed that it tended to leave discs where there was no water. By slowly pouring more water into the pan, the discs would shrink until there reach zero size, at which point a circular wave was emitted. I had just witnessed a topology change;) I could also reverse the process by taking some water out of the pan and stirring up the water. When the water had stabilised enough, discs were suddenly appearing and expanding quickly. That's my version of frying pan black holes: try it at home!

PS: the key is the circular wave.