As real manifolds, we can list various properties of the matrix groups for \({n>1}\) and \({rs\neq0}\).
| Group | Dimension | Compact | Connectedness |
|---|---|---|---|
| \({GL\left(n,\mathbb{R}\right)}\) | \({n^{2}}\) | No | 2 components |
| \({SL\left(n,\mathbb{R}\right)}\) | \({n^{2}-1}\) | No | Connected |
| \({O\left(n\right)}\) | \({n\left(n-1\right)/2}\) | Yes | 2 components |
| \({SO\left(n\right)}\) | \({n\left(n-1\right)/2}\) | Yes | Connected |
| \({O\left(r,s\right)}\) | \({n\left(n-1\right)/2}\) | No | 4 components |
| \({SO\left(r,s\right)}\) | \({n\left(n-1\right)/2}\) | No | 2 components |
| \({Sp\left(2n,\mathbb{R}\right)}\) | \({\left(2n\right)\left(2n+1\right)/2}\) | No | Connected |
| \({GL\left(n,\mathbb{C}\right)}\) | \({2n^{2}}\) | No | Connected |
| \({SL\left(n,\mathbb{C}\right)}\) | \({2(n^{2}-1)}\) | No | Simply connected |
| \({U\left(n\right)}\) | \({n^{2}}\) | Yes | Connected |
| \({SU\left(n\right)}\) | \({n^{2}-1}\) | Yes | Simply connected |
| \({Sp\left(2n,\mathbb{C}\right)}\) | \({\left(2n\right)\left(2n+1\right)}\) | No | Simply connected |
| \({Sp\left(n\right)}\) | \({n\left(2n+1\right)}\) | Yes | Simply connected |
Notes: In particular, since \({U(n)}\) is compact and connected, any unitary matrix \({U}\) can be written as \({U=e^{iH}}\) for some Hermitian matrix \({H}\).
Since the exponential map is surjective for any compact connected Lie group, it is surjective for \({SO(n)}\), \({U(n)}\) and \({SU(n)}\). As it turns out, it is also surjective for \({SO(3,1)^{e}}\) and \({GL(n,\mathbb{C})}\).
We can also characterize the topology of matrix groups by noting some diffeomorphisms of their manifolds:
- \({O(n+1)/O(n)\cong SO(n+1)/SO(n)\cong S^{n}}\)
- \({U(n+1)/U(n)\cong SU(n+1)/SU(n)\cong S^{2n+1}}\)
- In particular, we then have \({U(1)\cong SO(2)\cong S^{1},\: SU(2)\cong S^{3}}\)
- \({O(n)\cong S^{0}\times SO(n);\: U(n)\cong S^{1}\times SU(n)}\)
- Thus \({SO\left(n+1\right)/O(n)\cong\mathbb{R}\textrm{P}^{n}}\); \({SU\left(n+1\right)/U(n)\cong\mathbb{C}\textrm{P}^{n}}\)
- In particular, we then have \({SO(3)\cong\mathbb{R}\textrm{P}^{3}}\)
- \({SO(4)\cong S^{3}\times SO(3)}\); \({SO(8)\cong S^{7}\times SO(7)}\)
- \({Sp(2,\mathbb{R})\cong SL(2,\mathbb{R})}\); \({Sp(2,\mathbb{C})\cong SL(2,\mathbb{C})}\); \({Sp(1)\cong SU(2)\cong S^{3}}\)
With regard to homotopy groups, some facts are:
- \({\pi_{1}(G)}\) is abelian for any Lie group \({G}\) (in fact for any H-space)
- \({\pi_{2}(G)=0}\) for any Lie group \({G}\)
- \({\pi_{1}\left(SO\left(n\right)\right)=\mathbb{Z}_{2}}\) for \({n>2}\); \({\pi_{3}\left(SU\left(n\right)\right)=\mathbb{Z}}\) for \({n>1}\)