topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A real Lie group is compact if its underlying topological group is a compact topological group.
All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus $T$ of a compact Lie group is called the rank of $G$.
The normalizer $N(T)$ of a maximal torus $T$ determines $G$.
The Weyl group $W(G)=W(G,T)$ of $G$ with respect to a choice of a maximal torus $T$ is the group of automorphisms of $T$ which are restrictions of inner automorphisms of $G$.
The maximal torus is of finite index in its normalizer; the quotient $N(T)/T$ is isomorphic to $W(G)$.
The cardinality of $W(G)$ for a compact connected $G$, equals the Euler characteristic of the homogeneous space $G/T$ (“flag variety”).
See also at relation between compact Lie groups and reductive algebraic groups
The maximal torus of a connected compact Lie group is also the maximal abelian subgroup.
In particular:
All connected compact abelian Lie groups are tori, up to isomorphism.
(Classification of compact abelian Lie groups)
Assuming the axiom of choice, every abelian compact Lie group is isomorphic to the direct product group of an n-torus with a finite abelian group.
Write
for the subgroup which is the connected component of the neutral element in the given compact abelian Lie group $G$. By Prop. this is a torus
hence its underlying abelian group is a divisible group and therefore, by this Prop., an injective object in the category Ab of abelian group. This implies that the dashed extension in the following diagram in Ab exists:
hence that $G$ retracts onto $G_0$.
While this is, a priori, a diagram in abelian discrete groups not it abelian Lie group, the fact that the dashed morphism $p \colon G \to G_{\mathrm{e}}$ restricts to the identity morphism on $G_{\mathrm{e}}$, together with the assumption that $G$ is a disjoint union of copies of this connected component and using the homomorphism property implies that $p$ is a continuous homomorphism. But continuous homomorphisms of Lie groups are smooth, so that $p$ is in fact smooth.
Therefore we have a split exact sequence of Lie groups
and hence an isomorphism
By the assumption that $G$ was compact abelian, $A \,\coloneqq\, G/G_{\mathrm{e}}$ is finite abelian.
(compact Lie groups admit bi-invariant Riemannian metrics)
Every compact Lie group admits a bi-invariant Riemannian metric.
Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.
(e.g. Lee 12, Corollary 21.6)
The equivariant triangulation theorem (Illman 78, Illman 83) says that for $G$ a compact Lie group and $X$ a compact smooth manifold equipped with a smooth $G$-action, there exists a $G$-equivariant triangulation of $X$.
A discrete group is a compact Lie group iff it is a finite group.
The classical Lie groups for definite inner products are compact, such as the orthogonal groups, the unitary groups, the quaternionic unitary groups, etc., but not the Lorentz group etc.
Compact Lie groups make a somewhat unexpected appearance as equivariance groups in equivariant homotopy theory, where the compact Lie condition on the equivariance group is needed in order for (the available proofs of) the equivariant Whitehead theorem to hold.
(Namely, the equivariant triangulation theorem above is used in these proofs to guaratee that Cartesian products of coset spaces $G/H$ are themselves G-CW-complexes.)
In gauge theory (Yang-Mills theory/Chern-Simons theory, …) …
Textbook accounts:
Glen Bredon, Section 0.6 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)
Frank Adams, Lectures on Lie groups, University of Chicago Press, 1982 (ISBN:9780226005300, gbooks)
In the broader context of smooth manifolds:
Dedicated lecture notes:
Discussion in the context of representation theory:
For more see also the references at equivariant homotopy theory.
Last revised on November 4, 2021 at 18:05:36. See the history of this page for a list of all contributions to it.