Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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132 III Homology of Polyhedra<br />
all, be easily <strong>in</strong>terpreted, imag<strong>in</strong>e what might happen when we try to compute the<br />
homology of a more complex polyhedron. We now expla<strong>in</strong> what a “block” decomposition<br />
of a polyhedron is and how the homology of a polyhedron divided <strong>in</strong>to<br />
blocks is computed.<br />
Let K =(X,Φ) be a given simplicial complex; for every subset e ⊂ Φ, letebe the smallest subcomplex of K that conta<strong>in</strong>s e (we observe that e is not necessarily<br />
a simplicial complex; for <strong>in</strong>stance, take the previous triangulation T of the torus T 2<br />
and let e be the set of the simplexes {2}, {0,3,5}, and{3,4,6}; <strong>in</strong> this example,<br />
{0,3} ⊂{0,3,5} but {0,3} �∈ e). The simplicial complex e may be described <strong>in</strong><br />
another way:<br />
e = �<br />
σ .<br />
The simplicial complex e associated with the set e is called closure of e.<br />
For every e ⊂ Φ, wedef<strong>in</strong>etheboundary of e as<br />
σ∈e<br />
•<br />
e = e � e;<br />
this, by def<strong>in</strong>ition, is a subcomplex of K; furthermore, • e is a simplicial subcomplex<br />
of e:<br />
e � e ⊂ e ⇒ e � e ⊂ e = e ⇒ • e ⊂ e .<br />
F<strong>in</strong>ally, e = e ∪ • e: <strong>in</strong> fact, s<strong>in</strong>ce e ⊂ e and • e ⊂ e, we have the <strong>in</strong>clusion e ∪ • e ⊂ e; on<br />
the other hand, if σ ∈ e and σ �∈ e then σ ∈ e � e and thus<br />
σ ∈ e � e = • e ;<br />
we conclude that e ∪ • e ⊂ e.<br />
We extend the def<strong>in</strong>ition of closure of a set e ⊂ Φ to a cha<strong>in</strong> cn = ∑i miσ i n ∈<br />
Cn(K,Z): theclosure of cn is the simplicial complex<br />
σ<br />
i : mi�=0<br />
i n .<br />
cn = �<br />
(III.5.3) Def<strong>in</strong>ition. A p-block or block of dimension p of K is a set ep ⊂ Φ such<br />
that:<br />
(i) ep conta<strong>in</strong>s no simplex of dimension > p<br />
(ii) Hp(ep, • ep;Z) ∼ = Z<br />
(iii) (∀q �= p) Hq(ep, • ep;Z) ∼ = 0<br />
(III.5.4) Remark. To compute the homology of the pair of simplicial complexes<br />
(ep, • ep), it is necessary to give an orientation to K. However, we also could give an<br />
orientation to ep <strong>in</strong>dividually by choos<strong>in</strong>g a generator βp of the Abelian group<br />
Hp(ep, • ep;Z) ∼ = Zp(ep, • ep) ∼ = Z .