Simplicial Structures in Topology

III.3 Some Applications 119
we now consider the chain morphism (subdivision homomorphism)
ℵ: C(K) → C(K (1) )
defined in Theorem (III.2.2); we remind the reader that the homomorphism
C(g)ℵ r : C(K) −→ C(K)
is precisely the homomorphism that defines Hn( f ,Z) =Hn(C(g)ℵr n ). Since for
every n = 0,...,dimK,
(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q): Cn(K,Q) → Cn(K,Q)
is a linear transformation, we may define the number
dimK
Λ(C(g)ℵ)= ∑ (−1)
n=0
n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)] ;
it is natural to wonder whether there is any relation between this number and the
Lefschetz Number Λ( f ). The answer to this question is the next result, known as
the Hopf Trace Theorem.
(III.3.2) Lemma. For every map f : |K|→|K|, we have
Λ(C(g)ℵ)=Λ( f ) .
Proof. For every n ≥ 0, we consider two exact sequences
(1) 0 → Zn(K,Q) → Cn(K,Q) → Bn−1(K,Q) → 0
(2) 0 → Bn(K,Q) → Zn(K,Q) → Hn(K,Q) → 0
Then for any fixed t ∈ Z, we define the numbers
c(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)] ,
z(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)]|Zn(K,Q)) ,
b(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)]|Bn(K,Q)) ,and
h(t)=∑t n Tr (Hn( f ,Q)) .
By applying Lemma (III.3.1) to the exact sequences (1) and (2), we conclude that
c(t)=z(t)+tb(t)
z(t)=b(t)+h(t) ;
it follows from these two equalities that c(t) − h(t)=(1 + t)b(t); we arrive to the
thesis by setting t = −1. �
We now turn to the Lefschetz Fixed Point Theorem.