I really need to get a good introductory book on mathematical topology. Ideally one for laymen like "A Brief History of Time" did for physics, but I don't know if such a thing exists. Every time there's a story in the news about an advance in this field, it makes no sense at all to me. For example, the recent stories about the proof of the Poincare conjecture, and the subsequent refusal of the Fields Medal. From the first Times article about it:

To a topologist, a sphere, a cigar and a rabbit�s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.

Okay, certainly a strange way of looking at things, but so far I'm with you.

In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this "anything" had to be what mathematicians call compact, or closed, meaning that it has a finite extent.

Alright...

In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

When did we start talking about being able to shrink things without limit? And how do spheres and doughnuts exist in two dimensions? Or is it just the rubber band that's in two dimensions?

With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be.

What? Now I'm picturing a mathematician bent over an apple or a sponge, turning it over and over in his hands, shouting "Damn this conjecture! Where are the holes?" But evidently since we were talking about an apple as being two dimensional before, we're now really talking about four-dimensional objects:

...when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions.

And at this point I've pretty much checked out. Now I admit this may be an issue of the mainstream press's understanding and reporting of science as much as anything else. It's probable that a topologist would pick this account apart the way the guys at Language Log do any article that contains a single sentence or more that appears to make a claim related to linguistics. But I wouldn't have any chance at understanding the primary sources, the academic papers, so this is pretty much what I'm stuck with, unless anyone can show me a topology version of Language Log.

For quite a while I periodically puzzled over the Four Color Map problem, which stated (as I had heard it) that you never need more than four colors to have a map with no adjacent territories having the same color. I couldn't understand whether this referred to maps of the real world, or abstract maps with any conceivable layout of territories. Of course it would be odd for mathematicians to be concerned with the real world, but I couldn't see how it could be true in the abstract, because I assumed the real-world rule held that territories can be non-contiguous (imagine how many colors you would need on a world map that shaded embassies as part of the countries they represent). Finally a coworker informed me that they have to be contiguous, and that they have to share a side, not just a point. I then spent much of a day drawing shapes, trying unsuccessfully to find a counterexample, then being amazed when I realized it was true.