Simplicial Structures in Topology

III.5 Real Projective Spaces 139
We find a triangulation Mn−1 of RPn−1 in a similar way and so on. Note that
•
en coincides with the set of (n − 1)-simplexes en−1 of Mn−1 , which in turn is an
(n − 1)-block, etc. In this way, we obtain a block triangulation of RPn .
All that is left to do is to look into the generators of Hr(er, • er;Z) and their boundaries.
We denote a generator of Hn−1(Sn−1 ,Z) with zn−1 ∈ Zn−1(Sn−1 ;Z). 6 The
exact sequence of the homology groups of (Kn +,K n−1 ) shows that zn−1 ∗ an+1 is a
generating cycle of Hn(Kn + ,Kn−1 ;Z). It follows that ℵ(zn−1 ∗ an+1) is a generator
for Zn((Kn + )(1) ,(K n−1 ) (1) ;Z) and if q : |Kn + |→|Mn | is the quotient map, then
qℵ(zn−1 ∗ an+1) is a generator of Zn(|Mn |,|Mn−1 |;Z). Now,
∂qℵ(zn−1 ∗ an+1)=qℵ∂(zn−1 ∗ an+1)=
(−1) n qℵ(zn−1)=(−1) n qℵ(zn−2 ∗ an − zn−2 ∗ a ′ n) .
We interchange in zn−2 all points ar with a ′ r; this gives rise to an element which we
denote with z ′ n−2 ; indeed, zn−2 =(−1) n−1z ′ n−2 . Hence,
∂qℵ(zn−1 ∗ an+1)=(−1) n qℵ(zn−2 ∗ an − (−1) n−1 z ′ n−2 ∗ a′ n )
=(−1) n qℵ(zn−2 ∗ an) − (−1) n (−1) n−1 qℵ(z ′ n−2 ∗ a ′ n) .
Finally, since qℵ(zn−2 ∗ an+1)=qℵ(z ′ n−2 ∗ a′ n), we conclude that
∂qℵ(zn−1 ∗ an+1)=(1 +(−1) n )qℵ(zn−2 ∗ an) .
To simplify the notation, we write βr = qℵ(zr−1 ∗ ar+1); this is a generator of
Cr(e(M n );Z) and has the property
∂(βr)=(1 +(−1) r )βr−1 .
Specifically for 0 < 2r ≤ n, the group C2r(e(M n );Z) is generated by the elements
β2r and ∂(β2r) =2β2r−1; therefore, Z2r(e(Mn );Z) =0andH2r(RPn ;Z) =0. If
2r −1 < n, then the group C2r−1(e(Mn );Z) is generated by β2r−1 and ∂(β2r−1)=0;
hence, the group Z2r−1(e(Mn );Z) ∼ =Z is generated by β2r−1, the group B2r−1(e(Mn );
Z) is generated by 2β2r−1, and there is an isomorphism H2r−1(RPn ;Z) ∼ = Z2. We
have proved the following result:
(III.5.10) Theorem.
Hp(RP n ,Z) ∼ ⎧
⎪⎨
Z if p = 0
0 if 0 < p = 2q ≤ n
=
⎪⎩
Z2 if 0 < p = 2q − 1 < n
Z if n is odd and p = n .
6 We start with S 0 and select a generator z 0 = {a1}−{a ′ 1 };next,forS1 ,wedefinez1 as the join of
z0 and {a1,a ′ 1 },etc.