Simplicial Structures in Topology

176 V Triangulable Manifolds
are linearly independent cycles of Cn−1(S(x)) ⊗ Z2, because the (n − 2)-simplexes
of (dn−1 ⊗ 1Z2 )(cL) and (dn−1 ⊗ 1Z2 )(cM) appear always twice; consequently,
the vector space Hn−1(S(x);Z2) has dimension ≥ 2, a fact that contradicts the
well-known result Hn−1(S(x);Z2) ∼ = Z2 (see Theorem (II.5.5)). Therefore, M = /0
and K = L. �
Exercises
1. Show by means of an example that a closed subspace of an n-manifold is not
necessarily an n-manifold.
2. Let p be a positive integer, p ≥ 2, and let Cp be the (multiplicative) cyclic group
of order p of the p-th roots of the identity in C,
Cp = � ζ j p | j = 1...p� ,
where ζp = e2πi/p . If q is an integer prime to p, thenCpacts on S3 = {(z,w) ∈
C2 ||z| 2 + |w| 2 = 1}⊂C2 by setting
� � � j j
(z,w),ζ p ↦→ zζp,wζ qj�
p
for every j. Prove that the quotient space (called lens space and denoted by L(p,q))
is a 3-manifold.
3. Let G be a finite group that acts freely on a G-space X. Prove that if X/G is an
n-manifold, then also X is an n-manifold.
4. Prove that each compact manifold is homeomorphic to a subspace of an Euclidean
space R N ,forsomeN.
V.2 Closed Surfaces
In this section we study the compact connected topological 2-manifolds which we
shall henceforth call closed surfaces. In the previous section we gave some examples
of closed surfaces: S 2 , RP 2 , T 2 ∼ = S 1 ×S 1 . But there are more, like the complex
projective line (which, nevertheless and as we shall presently see, is homeomorphic
to the sphere S 2 ) and the Klein bottle. We begin by studying these examples in more
detail.
In order to construct the complex projective line CP 1 , we may proceed as follows:
we consider the space C 2 � {(0,0)} and the equivalence relation
(z0,z1) ≡ (z ′ 0,z ′ 1) ⇐⇒ (∃z ∈ C � {0}) z ′ 0 = zz0, z ′ 1 = zz1;