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