Simplicial Structures in Topology

118 III Homology of Polyhedra
linear transformation γU : U → U. Then γ induces a unique linear transformation
γW : W → W such that γW β = βγ, and therefore
Tr (γ)=Tr (γU)+Tr (γW ).
Proof. Let us consider a basis {�u1,...,�ur} of U and the vectors �vi = α(�ui) ∈ V,
i = 1,...,r. The latter are linearly independent in V and may, therefore, be completed
so that we have a basis {�v1,...,�vn} of V. Note that the vectors �wr+1 =
β (�vr+1),...,wn = β(�vn) constitute a basis for W. The matrix (aij), with i, j =
1,...,n, obtained by the equalities
γ(�vi)=
n
∑
j=1
a ji�v j , i = 1,...,n
represents γ. Transformations γU and γW are defined by the rules
and
γU(�ui)=
γW (�wi)=
r
∑
j=1
a ji�u j , for every i = 1,...,r
n
∑ a ji�w j , for every j = r + 1,...,n. �
j=r+1
We now concern ourselves with geometry. Our proofs will take place in the realm
of rational homology, that is to say, homology with coefficients in Q. We briefly
recall that, for every polyhedron |K|, Hn(|K|;Q) is a rational vector space of dimension
β (n). On the other hand, a map f : |K|→|K| produces a linear transformation
Hn( f ,Q): Hn(|K|;Q) → Hn(|K|;Q)
for each integer n ∈{0,...,dimK}. By definition, the Lefschetz Number of a map
f : |K|→|K| is the rational number
dimK
Λ( f )= ∑ (−1)
n=0
n Tr Hn( f ,Q) .
We consider a simplicial approximation g: K (r) → K to f (see Theorem (III.2.4))
that produces a chain morphism
and therefore, a homomorphism
C(g): C(K (r) ) → C(K)
Cn(g) ⊗ 1Q : C(K (r) ,Q) → C(K,Q) ;