Simplicial Structures in Topology

I.3 Group Actions 41
S2 = {(x,y,z) ∈ R3 | x2 + y2 + z2 = 1} by φ : S2 × Z2 → S2 , with φ((x,y,z),1) =
(x,y,z) and φ((x,y,z),−1) =(−x,−y,−z). Therefore, we obtain the orbit space
S2 /Z2 of this action by identifying antipodal points of the sphere S2 . On the other
hand, let E ∼ = S1 be the equator of S2 and let the antipodal points a and −a of S2 � E
be identified. Let S2 + be the north hemisphere of S2 ; we note that S2 + � E alone represents
the classes [a] ∈ S2 /Z2 where a ∈ S2 �E. In order to obtain S2 /Z ,wemust
2
is homeomorphic to the unit disk
still identify the antipodal points of E. SinceS 2 +
D2 , RP2 is also homeomorphic to the space S2 + /Z2 .
In general, Z2 acts on the n-dimensional unit sphere (n ≥ 1)
�
�
S n =
(x0,x1,...,xn) ∈ R n+1 |
by the antipodal action φ : S n × Z2 → S n , such that
and
n
∑
i=0
xi = 1
φ((x0,x1,...,xn),1)=(x0,x1,...,xn)
φ((x0,x1,...,xn),−1)=(−x0,−x1,...,−xn).
The orbit space Sn /Z2 is the n-dimensional real projective space .Whenn = 1, we
have the real projective line, which is homeomorphic to the circle S1 .
We say that a group G acts freely on a space X if
Exercises
(∀x ∈ X)(∀g ∈ G , g �= 1G) xg �= x.
1. Prove that the projective line RP 1 is homeomorphic to S 1 .
2. Prove that the action of a subgroup H ⊂ G of a topological group G given by the
product (g,h) ↦→ gh is an action (on the right) of H on G. Find the quotient G/H
when G = SO(2) and H ⊂ G is the group generated by a rotation angle θ.
3. Prove that the topological group GL(n,R) is connected.
4. Prove that all discrete subgroups of R are cyclic and infinite.
5. Find a (non-Abelian) subgroup of SO(3) that is free on two generators.
6. Consider the action of Q on R given by (x,q) ↦→ x+q. Is the quotient connected?
Hausdorff? Compact?
7. Let G be a topological group that acts on the left on two spaces X and Y .Prove
that the action of G
(g,[x ↦→ f (x)]) ∈ G × [X,Y] ↦→ [x ↦→ gf(g −1 x)] ∈ [X,Y ]
on the homotopy classes of maps is well defined.