Contents

group theory

# Contents

## Definition

A real Lie group is compact if its underlying topological group is a compact topological group.

## Properties

### Maximal tori

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”).

### Invariant metric

###### Proposition

(compact Lie groups admit bi-invariant Riemannian metrics)
Every compact Lie group admits a bi-invariant Riemannian metric.

### Smooth actions

###### Proposition

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)

### Equivariant triangulation theorem

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 $X$.

## Applications

### In equivariant homotopy theory

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.)