Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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III.5 Real Projective Spaces 133<br />
Note that a generator βp may be <strong>in</strong>terpreted as a l<strong>in</strong>ear comb<strong>in</strong>ation of p-simplexes<br />
of K or of ep; <strong>in</strong> any case, ∂(βp) is a cha<strong>in</strong> of • ep and therefore<br />
∂(βp) ⊂ • ep .<br />
(III.5.5) Def<strong>in</strong>ition. A block triangulation 5 of a simplicial complex K =(X,Φ) (or<br />
of a polyhedron |K|) isasete(K)={e i p} of blocks with the follow<strong>in</strong>g conditions:<br />
(1) For every σ ∈ Φ,thereisaunique block e(σ) of the set e(K) such that σ ∈ e(σ).<br />
(2) For every p-block ei p ∈ B(K), • ei p is a union of blocks with dimension < p.<br />
It follows from the preced<strong>in</strong>g def<strong>in</strong>ition that<br />
1. For every i, j, p, • e i p ∩ e j p = /0<br />
2. If i �= j, thene i p ∩ e j p = /0<br />
As an example, we give a block triangulation of the torus T 2 with the triangulation<br />
T previously described. We consider the follow<strong>in</strong>g sets of simplexes of T:<br />
1. e0 = {0}<br />
2. e1 1 = {{3},{4},{0,3},{3,4},{4,0}}<br />
3. e2 1 = {{1},{2},{0,1},{1,2},{2,0}}<br />
4. e2 = T � (e0 ∪ e1 1 ∪ e21 )<br />
We determ<strong>in</strong>e that the set<br />
e(T 2 )={e0,e 1 1 ,e2 1 ,e2}<br />
is a block triangulation for T 2 . We notice at once that<br />
e0 = {0} , • e0 = /0 ,<br />
e 1 1 = {0,3}∪{3,4}∪{4,0} , • e 1 1 = {0} ,<br />
e2 1 = {0,1}∪{1,2}∪{2,0} , • e 2 1 = {0} ,<br />
e2 = T , • e2 = e 1 1 ∪ e2 1 ;<br />
we now recall Theorem (III.4.1) and observe that<br />
|e i 1 |/| • e i 1 | ∼ = S 1 , i = 1,2 ,<br />
|e2|/| • e2| ∼ = S 2 ;<br />
this shows that the elements of e(T 2 ) are blocks. It is easily proved that they form a<br />
block triangulation.<br />
5 In spite of the name, this is not a triangulation. In fact, it is the analog of a cellular decomposition<br />
of |K|. The block homology can be seen as the cellular homology of such a cellular complex (see<br />
e.g. [7, Chap. V]).