So a while ago, I was reading Hatcher’s notes on 3-manifolds. In there, he defines what it means for a manifold to be prime and states, casually, that the 3-sphere is prime. He later says that it follows immediately from Alexander’s Theorem as, and I quote: Every 2-sphere in bounds a 3-ball. And that’s it. Done.
Elsewhere, Hatcher expands his above statement: …every 2-sphere in bounds a ball on each side…[and h]ence is prime. Again, though, it isn’t accompanied by anything, and while this is clearly a trivial result, I just couldn’t see it for the longest time…I knew that it followed from a number of things, e.g. the fact that is the identity of the connected sum operation, that is irreducible (and that every irreducible manifold is prime), that one gets the trivial sum by splitting along a 2-sphere in which bounds a 3-ball in , etc. Even so, I didn’t want to leverage some enormous machinery to deduce the smallest of results and what I really wanted was for someone to tell me what I was missing. So I never stopped thinking about this, even after moving forward, until finally – it just clicked!
I figure other people who are as visualization-impaired as I may benefit from seeing this explained in greater depth, so in lieu of typing a blog post containing something new and attention-worthy, I figure I’d share this instead. Details after the break.
Per my earlier entry: Given a smooth complex algebraic variety , I finally manage to track down a semi-manageable definition for the structure sheaf . But here’s the thing with super-abstract definitions of things:
There’s a big difference between “getting them” and understanding them. In this case, I read the definition a few dozen times, annotated the .pdf file by Elliott with what I found, and felt as if I truly “got it”. Then, I read an example regarding the topological space with the claim that …
…say what now?!
Finally, I reread a bunch of stuff and found a cool analogy between arbitrary varieties and the case where is a smooth -times continuously differentiable manifold of dimension . This cleared things up for me.
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos