manifolds, topology

Naturality of Flows

This is something I always forget exists and has a name, so I end up reproving it. Since this sequence of posts is a hodge-podge of things to help me take a differential geometry test, hopefully this will lodge the result in my brain and save me time if it comes up. I'm not sure… Continue reading Naturality of Flows

algebraic topology, manifolds, topology

Lie groups have abelian fundamental group

Last year I wrote up how to prove that the fundamental group of a (connected) topological group was abelian. Since Lie groups are topological groups, they also have abelian fundamental groups, but I think there is a much neater way to prove this fact using smooth things. Here it is: Lemma 1: A connected Lie… Continue reading Lie groups have abelian fundamental group

algebraic geometry, algebraic topology, manifolds

Cohomology of Abelian Varieties II

Two posts in the same week! Before we get started today, we need to introduce one more piece of new notation. Let $latex {\overline{T}}&fg=000000$ be the $latex {\mathbb{C}}&fg=000000$-antilinear maps $latex {V\rightarrow \mathbb{C}}&fg=000000$. Our goal is to prove that $latex {H^q(X, \mathcal{O}_X)\simeq \bigwedge^q\overline{T}}&fg=000000$ and $latex {H^q(X, \Omega^p)\simeq \bigwedge^pT\otimes \bigwedge^q\overline{T}}&fg=000000$. To do the calculation we will use… Continue reading Cohomology of Abelian Varieties II

algebraic geometry, algebraic topology, manifolds

Cohomology of Abelian Varieties

Hopefully I'll start updating more than once a month. Since it's been awhile and the previous post was tangent to what we're actually doing, I'll recap some notation. $latex {X}&fg=000000$ will be a compact complex (connected) Lie group of dimension $latex {g}&fg=000000$. We showed that we have an analytic isomorphism $latex {X\simeq (S^1)^{2g}\simeq (\mathbb{R}/\mathbb{Z})^{2g}}&fg=000000$. Let… Continue reading Cohomology of Abelian Varieties