I’ve been thinking for a while that I ought to start a research blog—something just to keep myself organized about the things I am thinking about, my thoughts on the papers I’ve read, my ideas, my questions. I figure I might as well make it public, though I seriously doubt anyone is going to read […]

Here’s a very easy theorem. Theorem. All closed orientable 3-manifolds are parallelizable. All closed orientable 3-manifolds are the boundary of a 4-manifold. Proof: Let be an orientable -manifold. Recall that the Wu class is the unique cohomology class such that , and Wu’s theorem says that . The up-shot is that Stiefel-Whitney classes are […]