Simplicial Structures in Topology

172 V Triangulable Manifolds
(V.1.2) Lemma. The Cartesian product of an n-manifold X by an m-manifold Y is
an (n + m)-manifold.
Proof. For any (x,y) ∈ X ×Y, we choose two charts (U,φ) and (V,ψ) of x ∈ X and
y ∈ Y , respectively; we note that (U ×V,φ × ψ) is a chart of the Cartesian product
of the manifolds: in fact, φ(U) × ψ(V) is an (elementary) open set of R n × R m ∼ =
R n+m . �
We now give some examples. It is easily proved that, for every n > 0, the Euclidean
space R n is an n-manifold. The circle S 1 ⊂ R, with the topology induced
by the Euclidean topology of R, is a 1-manifold; this is readily proved. The sphere
S 2 with the topology induced by R 3 is a 2-manifold (or surface): in fact, S 2 is
Hausdorff and its atlas is the set
A = {(S 2 � {−x},φx) | x ∈ S 2 }
where φx is the stereographic projection of S2 �{−x} from the point −x on the plane
Tx tangent to S2 at x. A similar result holds also for hyperspheres Sn , n > 2.
An immediate consequence from Lemma (V.1.2) is that the torus T 2 = S1 × S1 is a 2-manifold.
To prove that the real projective space RPn is an n-manifold, we use the following
theorem.
(V.1.3) Theorem. Let X be a compact n-manifold and let G be a finite topological
group freely acting on X. Then, the quotient space X/G is an n-manifold.
Proof. Let G be the finite group whose elements are g1 = 1G, g2,...,gp; then, the
orbit of any element x ∈ X consists in p (distinct) elements x = xg1,xg2,...,xgp.
For each pair (x,xgi), i = 2,...,p,wetakeapair(Ui,Vi) of open sets of X such that
x ∈ Ui, xgi ∈ Vi and Ui ∩Vi = /0.
Since Vi contains xgi, Vig −1
i surely contains x; therefore, the set
U =
p�
i=2
�
Ui ∩Vig −1
�
i
is an open set of X containing x, is disjoint from all open sets Vi, i = 2,...,p and,
consequently, from all Ugi, i = 2,...,p (because
Ugi ⊂ � Ui ∩Vig −1�
i gi ⊂ Vig −1
i gi = Vi
and Vi ∩U = /0 for each i = 2,...,p). The restriction to U of the canonical epimorphism
q: X → X/G,thatistosay,
q|U : U → q(U) ⊂ X/G