Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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128 III Homology of Polyhedra<br />
III.5 Homology of Real Projective Spaces<br />
We have already computed the homology groups of some elementary polyhedra<br />
such as the sphere S 2 and the torus T 2 (see also exercises <strong>in</strong> Sect. II.4). We have<br />
also studied techniques that, <strong>in</strong> theory, allow us to study the homology of polyhedra<br />
(for <strong>in</strong>stance, the Exact Homology Sequence Theorem). In this section, we compute<br />
the homology of real projective spaces; we shall see that the methods learned so far<br />
are not enough to complete our pre-established task and for this reason we shall<br />
provide a new method for comput<strong>in</strong>g homology groups.<br />
Those readers who have done the exercises of Sect. II.2 will have found at least<br />
one triangulation for RP 2 ; whatever the case, we consider the triangulation P of<br />
RP 2 that has 6 vertices, 15 edges, and 10 faces <strong>in</strong> Fig. III.3 (the geometric realization<br />
0<br />
2<br />
1<br />
3 4<br />
5<br />
1<br />
2<br />
0<br />
Fig. III.3<br />
of this simplicial complex is homeomorphic to the disk D2 1 with the identification of<br />
its boundary antipodal po<strong>in</strong>ts).<br />
In one of the exercises <strong>in</strong> Sect. II.4, we have asked the reader to compute the homology<br />
groups (with coefficients <strong>in</strong> Z) of a triangulation of RP2 ; the reader should<br />
have come to the results that are put together <strong>in</strong> the follow<strong>in</strong>g lemma (<strong>in</strong> any case,<br />
this lemma will be proved later on with another method).<br />
(III.5.1) Lemma.<br />
Hq(RP 2 ,Z) ∼ ⎧<br />
⎨ Z , q = 0<br />
= Z2 , q = 1<br />
⎩<br />
0 , q �= 0,1 .<br />
We now consider the n-dimensional real projective spaces RPn , with n ≥ 3. As<br />
for the two-dimensional case, RPn is a quotient space of Sn by identification of<br />
antipodal po<strong>in</strong>ts. Let us rephrase the def<strong>in</strong>ition of RPn as follows: We def<strong>in</strong>e the<br />
equivalence relation <strong>in</strong> Rn+1 � (0,...,0):<br />
x =(x0,...,xn) ≡ y =(y0,...,yn) ⇐⇒<br />
(∃λ ∈ R,λ �= 0)(∀i = 0,...,n)xi = λyi .