Simplicial Structures in Topology

132 III Homology of Polyhedra
all, be easily interpreted, imagine what might happen when we try to compute the
homology of a more complex polyhedron. We now explain what a “block” decomposition
of a polyhedron is and how the homology of a polyhedron divided into
blocks is computed.
Let K =(X,Φ) be a given simplicial complex; for every subset e ⊂ Φ, letebe the smallest subcomplex of K that contains e (we observe that e is not necessarily
a simplicial complex; for instance, take the previous triangulation T of the torus T 2
and let e be the set of the simplexes {2}, {0,3,5}, and{3,4,6}; in this example,
{0,3} ⊂{0,3,5} but {0,3} �∈ e). The simplicial complex e may be described in
another way:
e = �
σ .
The simplicial complex e associated with the set e is called closure of e.
For every e ⊂ Φ, wedefinetheboundary of e as
σ∈e
•
e = e � e;
this, by definition, is a subcomplex of K; furthermore, • e is a simplicial subcomplex
of e:
e � e ⊂ e ⇒ e � e ⊂ e = e ⇒ • e ⊂ e .
Finally, e = e ∪ • e: in fact, since e ⊂ e and • e ⊂ e, we have the inclusion e ∪ • e ⊂ e; on
the other hand, if σ ∈ e and σ �∈ e then σ ∈ e � e and thus
σ ∈ e � e = • e ;
we conclude that e ∪ • e ⊂ e.
We extend the definition of closure of a set e ⊂ Φ to a chain cn = ∑i miσ i n ∈
Cn(K,Z): theclosure of cn is the simplicial complex
σ
i : mi�=0
i n .
cn = �
(III.5.3) Definition. A p-block or block of dimension p of K is a set ep ⊂ Φ such
that:
(i) ep contains no simplex of dimension > p
(ii) Hp(ep, • ep;Z) ∼ = Z
(iii) (∀q �= p) Hq(ep, • ep;Z) ∼ = 0
(III.5.4) Remark. To compute the homology of the pair of simplicial complexes
(ep, • ep), it is necessary to give an orientation to K. However, we also could give an
orientation to ep individually by choosing a generator βp of the Abelian group
Hp(ep, • ep;Z) ∼ = Zp(ep, • ep) ∼ = Z .