Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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III.3 Some Applications 119<br />
we now consider the cha<strong>in</strong> morphism (subdivision homomorphism)<br />
ℵ: C(K) → C(K (1) )<br />
def<strong>in</strong>ed <strong>in</strong> Theorem (III.2.2); we rem<strong>in</strong>d the reader that the homomorphism<br />
C(g)ℵ r : C(K) −→ C(K)<br />
is precisely the homomorphism that def<strong>in</strong>es Hn( f ,Z) =Hn(C(g)ℵr n ). S<strong>in</strong>ce for<br />
every n = 0,...,dimK,<br />
(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q): Cn(K,Q) → Cn(K,Q)<br />
is a l<strong>in</strong>ear transformation, we may def<strong>in</strong>e the number<br />
dimK<br />
Λ(C(g)ℵ)= ∑ (−1)<br />
n=0<br />
n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)] ;<br />
it is natural to wonder whether there is any relation between this number and the<br />
Lefschetz Number Λ( f ). The answer to this question is the next result, known as<br />
the Hopf Trace Theorem.<br />
(III.3.2) Lemma. For every map f : |K|→|K|, we have<br />
Λ(C(g)ℵ)=Λ( f ) .<br />
Proof. For every n ≥ 0, we consider two exact sequences<br />
(1) 0 → Zn(K,Q) → Cn(K,Q) → Bn−1(K,Q) → 0<br />
(2) 0 → Bn(K,Q) → Zn(K,Q) → Hn(K,Q) → 0<br />
Then for any fixed t ∈ Z, we def<strong>in</strong>e the numbers<br />
c(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)] ,<br />
z(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)]|Zn(K,Q)) ,<br />
b(t)=∑t n Tr [(Cn(g) ⊗ 1Q)(ℵ r n ⊗ 1Q)]|Bn(K,Q)) ,and<br />
h(t)=∑t n Tr (Hn( f ,Q)) .<br />
By apply<strong>in</strong>g Lemma (III.3.1) to the exact sequences (1) and (2), we conclude that<br />
c(t)=z(t)+tb(t)<br />
z(t)=b(t)+h(t) ;<br />
it follows from these two equalities that c(t) − h(t)=(1 + t)b(t); we arrive to the<br />
thesis by sett<strong>in</strong>g t = −1. �<br />
We now turn to the Lefschetz Fixed Po<strong>in</strong>t Theorem.