4
votes
2answers
57 views

$\pi_n(SU(2)/Z_N)\simeq?$, $\pi_n(SO(3)/Z_N)\simeq?$, $\pi_n(U(1)/Z_N)\simeq?$

So based on the tool, I have attempted to compute the following threes homotopy groups. $\pi_n(SU(2)/Z_N)\simeq?$ $\pi_n(SO(3)/Z_N)\simeq?$ $\pi_n(U(1)/Z_N)\simeq?$ With a fixed positive integer ...
5
votes
2answers
173 views

Homotopy groups of some magnetic monopoles

This is a list of homotopy groups which I (as a physics researcher) encounter when studying magnetic monopole under certain configuration of gauge field profiles. \begin{gather} \pi_2(SU(2)/U(1)) ...
5
votes
0answers
84 views

$\pi_2(G)$ for $G$ a Lie group. [duplicate]

It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory? Also, who gave the first proof of ...
3
votes
1answer
207 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
60 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
42 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
71 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
269 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
77 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 ...