Simplicial Structures in Topology

III.4 Relative Homology 125
6. Let M + 3×3 be the set of all real square matrices with no negative elements. Apply
the preceeding exercise to prove that every matrix A ∈ M + 3×3 has at least one nonnegative
eigenvalue.
III.4 Relative Homology for Polyhedra
In Sect. II.4, we have studied the relative homology of a pair of simplicial complexes
(K,L); specifically, we have proved that given two simplicial complexes K =(X,Φ)
and L =(Y,Θ) with Y ⊂ X and Θ ⊂ Φ, then
(∀n ≥ 1) Hn(K,L;Z) ∼ = Hn(K ∪CL;Z)
where CL is the abstract cone vL Theorem (II.4.7). In Sect. III.2, wehavestudied
the homology functor in the category of polyhedra, giving the definition of
H∗(|K|,Z) := H(K,Z) for any polyhedron |K| (the definition of functor H∗(−,Z)
on morphisms is more intricate and depends on the Simplicial Approximation Theorem).
Nevertheless, we may say that
(∀n ≥ 1) Hn(|K|,|L|;Z) ∼ = Hn(|K ∪CL|;Z)
for every pair of polyhedra (|K|,|L|). In this section, we prove the following result:
(III.4.1) Theorem. For every pair of polyhedra (|K|,|L|) (with L a subcomplex of
K) and every integer n ≥ 1,
Hn(|K|,|L|;Z) ∼ = Hn(|K|/|L|;Z) .
This theorem is a direct consequence of a more general result in the category of
topological spaces:
(III.4.2) Theorem. Let (X,A) be a pair of topological spaces with A closed in X;
suppose that (X,A) has the Homotopy Extension Property. Let CA be the cone
(A × I)/(A ×{0}). Then the adjunction space X ∪CA and the quotient space X/A
are of the same homotopy type.
Proof. The reader may intuitively come to this result by considering the cone CA as
a space contractible to a point; here is a rigorous proof of the statement.
In this theorem, we have two pushouts: one for constructing X ∪CA and the other
for X/A; these are their corresponding diagrams:
i1
A
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CA
i ��
i
i1
X
��
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X ∪CA