# Manifold properties of matrix groups

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 $${n>1}$$