Simplicial Structures in Topology

I.3 Group Actions 39
I.3 Group Actions
A topological group is a group together with a topology such that the functions
defined by the multiplication and by inverting the elements of the group are continuous;
in more precise terms, let G be a group with the multiplication
×: G × G → G , (∀(g,g ′ ) ∈ G × G)(g,g ′ ) ↦→ gg ′ ;
let us consider G × G with the product topology; we then request that the two maps
(g,g ′ ) ↦→ gg ′ and g ↦→ g −1 be continuous functions.
Here are some examples of topological groups; the details of the proofs are left
to the reader.
1. All groups with discrete topology.
2. The additive group R with the Euclidean topology (given by the distance
(∀x,y ∈ R) d(x,y)=|x − y|).
3. The multiplicative group R ∗ = R�{0} with the topology given in the previous
example.
4. The additive group C of the complex numbers with the topology given by the
distance
(∀x,y ∈ C) d(x,y)=|x − y| .
5. Let GL(n,R) be the multiplicative group of all real, invertible square matrices
of rank n (the general linear group). We define a topology on GL(n,R) as
follows: we note that the function
(aij)i, j=1,...,n ∈ M(n,R) ↦→ (a11,a12,...,a21,a22,...,ann) ∈ R n2
from the set M(n,R) of all real matrices n × n to the set Rn2 is a bijection.
This function defines a topology on M(n,R), which derives from the Euclidean
topology on Rn2 and induces a topology on GL(n,R). With this topology, the
group GL(n,R) becomes a topological group.
6. Let SO(n) be the subgroup of GL(n,R) of the matrices M ∈ GL(n,R) such that
M−1 is the transpose of M and detM = 1; the previously defined topology on
GL(n,R) induces a topological group structure on SO(n).
Let G be a topological group with neutral element 1G and X be a topological
space. An action (on the right) of G on X is a continuous function
such that
φ : X × G → X
(a) (∀x ∈ X) φ(x,1G)=x
(b) (∀x ∈ X)(∀g,g ′ ∈ G) φ(φ(x,g),g ′ )=φ(x,gg ′ ).
We say that G acts on the right of X (through the action φ). To make it simple,
we write φ(x,g)=xg. Similarly, it is possible to define an action on the left.