Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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III.5 Real Projective Spaces 139<br />
We f<strong>in</strong>d a triangulation Mn−1 of RPn−1 <strong>in</strong> a similar way and so on. Note that<br />
•<br />
en co<strong>in</strong>cides with the set of (n − 1)-simplexes en−1 of Mn−1 , which <strong>in</strong> turn is an<br />
(n − 1)-block, etc. In this way, we obta<strong>in</strong> a block triangulation of RPn .<br />
All that is left to do is to look <strong>in</strong>to the generators of Hr(er, • er;Z) and their boundaries.<br />
We denote a generator of Hn−1(Sn−1 ,Z) with zn−1 ∈ Zn−1(Sn−1 ;Z). 6 The<br />
exact sequence of the homology groups of (Kn +,K n−1 ) shows that zn−1 ∗ an+1 is a<br />
generat<strong>in</strong>g cycle of Hn(Kn + ,Kn−1 ;Z). It follows that ℵ(zn−1 ∗ an+1) is a generator<br />
for Zn((Kn + )(1) ,(K n−1 ) (1) ;Z) and if q : |Kn + |→|Mn | is the quotient map, then<br />
qℵ(zn−1 ∗ an+1) is a generator of Zn(|Mn |,|Mn−1 |;Z). Now,<br />
∂qℵ(zn−1 ∗ an+1)=qℵ∂(zn−1 ∗ an+1)=<br />
(−1) n qℵ(zn−1)=(−1) n qℵ(zn−2 ∗ an − zn−2 ∗ a ′ n) .<br />
We <strong>in</strong>terchange <strong>in</strong> zn−2 all po<strong>in</strong>ts ar with a ′ r; this gives rise to an element which we<br />
denote with z ′ n−2 ; <strong>in</strong>deed, zn−2 =(−1) n−1z ′ n−2 . Hence,<br />
∂qℵ(zn−1 ∗ an+1)=(−1) n qℵ(zn−2 ∗ an − (−1) n−1 z ′ n−2 ∗ a′ n )<br />
=(−1) n qℵ(zn−2 ∗ an) − (−1) n (−1) n−1 qℵ(z ′ n−2 ∗ a ′ n) .<br />
F<strong>in</strong>ally, s<strong>in</strong>ce qℵ(zn−2 ∗ an+1)=qℵ(z ′ n−2 ∗ a′ n), we conclude that<br />
∂qℵ(zn−1 ∗ an+1)=(1 +(−1) n )qℵ(zn−2 ∗ an) .<br />
To simplify the notation, we write βr = qℵ(zr−1 ∗ ar+1); this is a generator of<br />
Cr(e(M n );Z) and has the property<br />
∂(βr)=(1 +(−1) r )βr−1 .<br />
Specifically for 0 < 2r ≤ n, the group C2r(e(M n );Z) is generated by the elements<br />
β2r and ∂(β2r) =2β2r−1; therefore, Z2r(e(Mn );Z) =0andH2r(RPn ;Z) =0. If<br />
2r −1 < n, then the group C2r−1(e(Mn );Z) is generated by β2r−1 and ∂(β2r−1)=0;<br />
hence, the group Z2r−1(e(Mn );Z) ∼ =Z is generated by β2r−1, the group B2r−1(e(Mn );<br />
Z) is generated by 2β2r−1, and there is an isomorphism H2r−1(RPn ;Z) ∼ = Z2. We<br />
have proved the follow<strong>in</strong>g result:<br />
(III.5.10) Theorem.<br />
Hp(RP n ,Z) ∼ ⎧<br />
⎪⎨<br />
Z if p = 0<br />
0 if 0 < p = 2q ≤ n<br />
=<br />
⎪⎩<br />
Z2 if 0 < p = 2q − 1 < n<br />
Z if n is odd and p = n .<br />
6 We start with S 0 and select a generator z 0 = {a1}−{a ′ 1 };next,forS1 ,wedef<strong>in</strong>ez1 as the jo<strong>in</strong> of<br />
z0 and {a1,a ′ 1 },etc.