0
votes
1answer
247 views

Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
5
votes
4answers
703 views

Is a measurable homomorphism on a Lie group smooth?

Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth? Edit: My original question said "measurable ...
6
votes
1answer
518 views

G-equivariant Whitehead's Theorem

Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...
2
votes
1answer
456 views

Meaning of “Compact” in 1932 Paper by van der Waerden “Continuity Theorem for Semisimple Lie Groups”.

I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions. I am attempting ...
5
votes
3answers
658 views

On closed totally disconnected subgroups of connected real Lie groups

So the following statement seems to be obvious but I don't see how to prove it: Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete? Note that ...
0
votes
2answers
579 views

Looking for general approaches to show connectedness of topological groups

Let $G$ be a topological group. One general approach to show that $G$ is connected is the following: For every subgroup $H\leq G$ (not necessarily closed) we have a projection map: $$ \pi: ...
9
votes
2answers
647 views

Is there a way to see a topological group as the “Cayley graph” of its “infinitesimal generators”?

At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
6
votes
3answers
1k views

Using topology to characterize embedded Lie subgroups of Lie groups.

Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup. This leads us to ask the following question: Can we replace "topologically closed" with ...
19
votes
6answers
932 views

Explanation for E_8's torsion

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...