Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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138 III Homology of Polyhedra<br />
III.5.2 Homology of RP n , with n ≥ 4<br />
We first wish to f<strong>in</strong>d a convenient triangulation for RP n . We rem<strong>in</strong>d the reader that<br />
RP n is also <strong>in</strong>terpreted as the pushout space of the diagram<br />
ın−1<br />
S n−1<br />
��<br />
D n<br />
qn−1 ��<br />
RP n−1<br />
where qn−1 is the map that identifies antipodal po<strong>in</strong>ts of S n−1 and ın−1 is the <strong>in</strong>clusion.<br />
In fact, <strong>in</strong> this way we have a sequence of real projective spaces<br />
RP 1 ⊂ ...⊂ RP n−1 ⊂ RP n .<br />
Let Sn + be the northern hemisphere of Sn , that is to say, the set of all po<strong>in</strong>ts<br />
(x1,...,xn+1) ∈ Rn+1 �<br />
,wherer = ∑ n+1<br />
i=1 x2 i = 1andxn+1≥0. We def<strong>in</strong>e the function<br />
f : Dn → Sn + as follows:<br />
�<br />
(<br />
f (x1,...,xn)=<br />
x1<br />
�<br />
πr xn πr πr<br />
r s<strong>in</strong> 2 ,..., r s<strong>in</strong> 2 ,cos 2 ) if r = ∑ n i=1 x2 i �= 0<br />
(0,0,...,1) if r = 0 .<br />
This function is a homeomorphism whose restriction to the boundary of S n + is the<br />
identity. With this <strong>in</strong> m<strong>in</strong>d, we may say that RPn is obta<strong>in</strong>ed from Sn + by identify<strong>in</strong>g<br />
the antipodal po<strong>in</strong>ts of ∂Sn + ∼ = Sn−1 .<br />
Let Kn−1 be the standard triangulation of Sn−1 (see p. 55). We def<strong>in</strong>e a triangulation<br />
Kn + of Sn + as the jo<strong>in</strong> of Kn−1 and {an+1}, wherean+1 =(0,...,0,1) ∈ Rn+1 .<br />
We are tempted to def<strong>in</strong>e a triangulation for RPn by identify<strong>in</strong>g the antipodal vertices<br />
of Sn−1 ; it does not work, as we could have different simplexes def<strong>in</strong>ed by the<br />
same vertices <strong>in</strong> RPn . The trick is to work with barycentric subdivisions. Hence, we<br />
def<strong>in</strong>e the triangulation Mn of RPn , identify<strong>in</strong>g the antipodal vertices of (Kn−1 ) (1) <strong>in</strong><br />
(Kn + )(1) .Thesetenof n-simplexes of Mn is an n-block of Mn : <strong>in</strong> fact, the simplicial<br />
function ((Kn + )(1) ,(K n−1 ) (1) ) → (en, • en) is <strong>in</strong>jective <strong>in</strong>to (Kn + )(1) �(K n−1 ) (1) and <strong>in</strong>duces<br />
therefore an isomorphism among the cha<strong>in</strong> complexes C((Kn + )(1) ,(K n−1 ) (1) ),<br />
and C(en, • en); on the other hand, the relative version of the cha<strong>in</strong> complexes homomorphism<br />
ℵ : C((K n + ),(Kn−1 );Z) → C(K n + )(1) ,(K n−1 ) (1) ;Z) ,<br />
def<strong>in</strong>ed <strong>in</strong> Theorem (III.2.2), <strong>in</strong>duces an isomorphism between the relative homology<br />
of pairs ((K n + )(1) ,(K n−1 ) (1) )) and (en, • en); hence,<br />
Hr(en, • en;Z) ∼ =<br />
�<br />
Z if r = n<br />
0 ifr �= n.