It is after 5pm Pacific Time, so at last my lips are unsealed...I can reveal to you at last the shocking, explosively revelatory news that the universal covering space of RP²vRP² looks like an infinitely long caterpillar, or snowman, or string of anal beads. To see why this should be, imagine someone hands you a wedge of two projective planes, and asks you to cover them. A projective plane is just a crushed sphere, so the first thing you'd probably think to do is to blow up one of your planes into a sphere again. You want to build a two-sheeted covering of your original space; your blown-up sphere covers one of the planes twice, but the other plane is covered only once so far, by itself, so to balance things out we need to attach another copy of the projective plane to the sphere antipodally from the first. Then you'd say to yourself: Well, Self, the obvious thing to do now is to blow up one of these projective planes again...You'll get a sphere there, connected at one end to another sphere, which is connected at its other end to a projective plane; and now, to keep the covering even, you have to attach both another sphere and then a projective plane to your new sphere, antipodally, so you always get a symmetrical structure...If you keep blowing your planes up indefinitely, you'll wind up with the Snowman Space, which turns out to be simply connected and a countable cover of the original space. Keen, eh?

This take-home Manifolds final was my only exam. I love graduate school. I'm now free of all responsibilities 'til January. And you know what that means: video games that involve smashing dirty zombies with a pipe.