I’m not exactly sure what to tell you. If only I were a better stand-up comic, I could put something together. Maybe I will just say a word about the PL versus the differential stuff. I think it’s always been a kind of a two-sided thing here that even very long ago when Poincaré was committing his duality theorem, for instance, he first of all defined Betti numbers using… it’s not really clear what he was using, but I think he was using smooth things. And then he decided to make a more precise definition and used his triangulations and dual cells and stuff like that, so that at that point you might say he saw that there was some kind of logical difficulty in the smooth case. Anyhow, maybe not a real logical difficulty, but somehow a difficulty, and he got this PL version of things. And then at the same time he also more or less invented differential forms, which means no triangulations at all, and brought it back to the differential world. So I think it’s a matter of the interaction of these things, it’s a matter of taste. For example, my idea is that the reason I’ve never been really happy with the real numbers is that I must have had a very bad experience as a child and not that good a time in freshman calculus. I personally have a preference even for just incredibly horrible objects, or finite things. But this is obviously just a matter of preference. Anyway, that’s all for my stand-up comedy.
KeywordsBetti Number Morse Function Differential Structure Differentiable Structure Homotopy Sphere
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