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