Simplicial Structures in Topology

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