see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
Theorems
For $G$ a topological group a compact subgroup is a topological subgroup $K \subset G$ which is a compact group.
A compact subgroup $K \hookrightarrow G$ is called maximal compact if every compact subgroup of $G$ is conjugate to a subgroup of $K$.
If it exists then, by definition, it is unique up to conjugation.
A locally compact topological group $G$ is called almost connected if the quotient topological space $G/G_0$ (of $G$ by the connected component of the neutral element) is compact.
See for instance (Hofmann-Morris, def. 4.24).
Every compact and every connected topological group is almost connected.
Also every quotient of an almost connected group is almost connected.
Let $G$ be a locally compact almost connected topological group.
Then
$G$ has a maximal compact subgroup $K$;
the coset space $G/K$ is homeomorphic to a Euclidean space.
This is due to (Malcev) and (Iwasawa). See for instance (Stroppel, theorem 32.5).
Let $G$ be a locally compact almost connected topological group.
Then a compact subgroup $K \hookrightarrow G$ is maximal compact precisely if the coset space $G/K$ is contractible
(in which case, due to theorem 1, it is necessarily homeomorphic to a Euclidean space).
This is (Antonyan, theorem 1.2).
In particular, in the above situation the subgroup inclusion
is a homotopy equivalence of topological spaces.
The following table lists some Lie groups and their maximal compact Lie subgroups. See also compact Lie group.
Lie group | maximal compact subgroup |
---|---|
real general linear group $GL(n, \mathbb{R})$ | orthogonal group $O(n)$ |
its connected component $GL(n,\mathbb{R})_0$ | special orthogonal group $SO(n)$ |
complex general linear group $GL(n, \mathbb{C})$ | unitary group $U(n)$ |
complex special linear group $SL(n, \mathbb{C})$ | special unitary group $SU(n)$ |
symplectic group $Sp(2n,\mathbb{R})$ | unitary group $U(n)$ |
complex symplectic group? $Sp(2n,\mathbb{C})$ | compact symplectic group $Sp(n)$ |
Narain group $O(n,n)$ | two copies of the orthogonal group $O(n) \times O(n)$ |
unitary group $U(p,q)$ | $U(p) \times U(q)$ |
special Lorentz/AdS etc. group $SO(p,q)$ | $SO(p) \times SO(q)$ |
Lorentz / AdS spin group $Spin(q,p)$ | $Spin(q) \times Spin(q) / \{(1,1), (-1,-1)\}$ |
The following table lists specifically the maximal compact subgroups of the “$E$-series” of Lie groups culminating in the exceptional Lie groups $E_n$.
$n$ | real form $E_{n(n)}$ | maximal compact subgroup $H_n$ | $dim(E_{n(n)})$ | $dim(E_{n(n)}/H_n )$ |
---|---|---|---|---|
2 | $SL(2, \mathbb{R}) \times \mathbb{R}$ | $SO(2)$ | 4 | 3 |
3 | $SL(3,\mathbb{R}) \times SL(2,\mathbb{R})$ | $SO(3) \times SO(2)$ | 11 | 7 |
4 | $SL(5, \mathbb{R})$ | $SO(5)$ | 24 | 14 |
5 | $Spin(5,5)$ | $(Sp(2) \times Sp(2))/\mathbb{Z}_2$ | 45 | 25 |
6 | E6(6) | $Sp(4)/\mathbb{Z}_2$ | 78 | 42 |
7 | E7(7) | $SU(8)/\mathbb{Z}_2$ | 133 | 70 |
8 | E8(8) | $Spin(16)/\mathbb{Z}_2$ | 248 | 128 |
A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the Prüfer group $\mathbb{Z}[1/p]/\mathbb{Z}$ endowed with the discrete ($0$-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.
A general survey is given in
Textbooks with relevant material include
M. Stroppel, Locally compact groups, European Math. Soc., (2006)
Karl Hofmann, Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)
Original articles include
M. Peyrovian, Maximal compact normal subgroups, Proceedings of the American Mathematical Society, Vol. 99, No. 2, (1987) (jstor)
Sergey A. Antonyan, Characterizing maximal compact subgroups (arXiv:1104.1820v1)