Simplicial Structures in Topology

130 III Homology of Polyhedra
such that
ℓqn ¯ = gn and ℓīn = jn .
We wish to prove that the map ℓ is a homeomorphism. We first prove that ℓ is
bijective; to this end, it is sufficient to prove that the restriction
ℓ : D n+1 � S n −→ RP n+1 � RP n
is bijective. With this in mind, we define the function
˜gn : RP n+1 � RP n −→ D n+1 � S n
such that, for every [(x0,...,xn+1)] ∈ RP n+1 � RP n ,
where
˜gn([(x0,...,xn+1)]) =
r =
�
x0xn+1 xnxn+1
,...,
|xn+1|r |xn+1|r
�
n+1
∑ |xi|
0
2 ,
and we notice that gn ˜gn = ˜gngn = 1. The continuity of the inverse function ℓ −1
derives from the fact that ℓ is a bijection from the compact space RP n ⊔qn Dn+1 onto
the Hausdorff space RP n+1 (see Theorem (I.1.27)).
(III.5.2) Proposition.
Hq(RP 3 ,Z) ∼ ⎧
Z if q = 0
⎪⎨
Z2 if q = 1
=
⎪⎩
Z if q = 3
0 if q �= 0,1,3.
Proof. The exact homology sequence for the pair (RP 3 ,RP 2 ) is
... ��
Hn(RP 2 ;Z) Hn(i)
��
Hn(RP 3 ;Z) q∗(n)
��
Hn(RP 3 ,RP 2 ;Z) λn ��
λn ��
Hn−1(RP 2 ;Z)
��
Hn−1(RP 3 ;Z)
�
��
...
(see Theorem (II.4.1)). By Lemma (III.5.1), Hn(RP 2 ;Z) is zero if n �= 0,1. By
Theorem (II.4.7) and considering that
RP 3 /RP 2 ∼ = S 3 ,
we have that, for every n ≥ 1, Hn(RP 3 ,RP 2 ;Z) ∼ = Hn(S 3 ) is zero if n �= 3andis
isomorphic to Z if n = 3. The significant part of the sequence becomes