Simplicial Structures in Topology

III.3 Some Applications 117
where HP is the category of polyhedra and of homotopy classes of maps among
polyhedra.
(III.2.10) Remark. Corollary (III.2.9) tells us in particular that, for every polyhedron
|K| and whatever barycentric subdivision |K (r) |,wehave
H∗(|K|;Z) ∼ = H∗(|K (r) |;Z);
moreover, the homology H∗(X;Z) of a compact and triangulable topological space
is invariant under the chosen triangulation of X.
Exercises
1. Prove that a projection π : K (1) → K is a simplicial approximation of the identity
map 1: |K|→|K|.
2. Prove (using the Simplicial Approximation Theorem) that the set [|K|,|L|] of
homotopy classes of maps from a polyhedron |K| to a polyhedron |L| is countable.
III.3 Some Applications
In this section, we give some applications of simplicial homology to geometry. Our
first important result is the Lefschetz Fixed Point Theorem. Before we state it, let
us review some well-known results in linear algebra.
The trace of a rational square matrix (aij), i = j = 1,...,n is the rational number
Tr ((aij)) =
n
∑
i=1
aii .
We note that for two rational square matrices A and B,
Tr (AB)=Tr (BA) .
Thus, we may define the trace of a linear transformation α : V → V of an
n-dimensional rational vector space. Indeed, if A represents α, thenB represents
α if and only if there exists C such that B = CAC −1 . We then define
(III.3.1) Lemma. Let
Tr (α)=Tr (B)=Tr (A) .
0 → U α
−→ V β
−→ W → 0
be an exact sequence of finite dimensional rational vector spaces and of linear transformations.
Let γ : V → V be a linear transformation whose restriction to U is a