Simplicial Structures in Topology

III.5 Real Projective Spaces 135
Let e i q be any q-block (recall that q > p). We begin by removing cp from e i q:we
write
cp = c i p + f i p where c i p ⊂ e i q , f i p ⊂ e (q) � e i q .
Hence, ∂p(c i p) ⊂ e i q; in addition,
which leads to
∂p(c i p) ⊂ ∂p(cp) ∪ ∂p( f i p) ⊂ e (p−1) ∪ (e (q) � e i q)=e (q) � e i q
∂p(c i p ) ⊂ ei q ∩ (e(q) � e i q )= • e i q .
Therefore, c i p ∈ Zp(e i q , • e i p );sincep �= q,wehaveHp(e i q , • e i p )=0andso
c i p = ∂p+1( f i p+1 )+gip , where f i p+1 ∈ Cp+1(ei q ) and gip ∈ Cp( • e i q ) .
Hence, cp is homologous to
cp − ∂p+1( f i p+1 )=ci p + f i p − (ci p − gi p )= f i p + gi p ;
note that the closure of the p-chain f i p + gip is contained in
�
e (q) � e i �
p ∪ • e i q = e (q) � e i q ;
since f i p + gi p is homologous to cp, we may say that we have removed cp from the
block e i p.
Let us now remember that i �= j =⇒ e i q ∩e j q = /0; then, because ∂p+1( f i p+1 ) ⊂ ei q,
the coefficient of each p-simplex of ∂p+1( f i p+1 ) in e j q is zero. Hence for every i, we
may find a (p + 1)-chain f i p+1 such that
cp −∑ i
∂p+1( f i p+1 ) ⊂ e(q−1) .
We define the p-chain fp = cp − ∑i ∂p+1( f i p+1 ); from the observations made at the
beginning of the proof, we conclude that there is a p-chain c ′ p of K homologous to
cp andsuchthat∂p(c ′ p ) ⊂ e(p) .
Now the proof proceeds as before: let ei p be an arbitrary p-block; we write
c ′ p = ki p + hi p where ki p ⊂ ei p , hi p ⊂ e(p) � e i p and hi p ⊂ e(p) � e i p .
As before, ki p ∈ Zp(ei p , • ei p ).However,Zp(ei p , • ei p ) ∼ = Z;letβi p be one of its generators;
then, ki p = miβ i p for a certain nonzero integer mi. Repeating this procedure for each
i we obtain a chain of p-blocks with
c ′ p −∑ i
miβ i p ⊂ e(p−1) ;