Tagged Questions
3
votes
1answer
77 views
Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?
I'm looking for a reference for the following result:
If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
2
votes
0answers
41 views
Equivariant homotopy equivalence of based loop group
Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
3
votes
0answers
28 views
Fundamental group of $Spin^c(2)$?
Is the fundamental group of $Spin^c(2)$, the second complex spin group, also $\mathbb{Z}$?
If so, how does one see this?
Just to avoid any confusion, my definition is:
$$Spin^c(2) = (SO(2) \times ...
1
vote
1answer
63 views
Connection between spaces of cosets and homotopy cofibers
Suppose we have an inclusion of a closed sub-Lie group $H\to G$ and take the space of left cosets of $H$, $G/H$. How is this related to the homotopy cofiber of the inclusion of topological spaces ...
15
votes
1answer
185 views
Geometric interpretation of the map $SO(4) \to SO(3)$
Let me first explain the background of my question.
As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
2
votes
0answers
46 views
How to prove that a lie group is simply connected
I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...
