Simplicial Structures in Topology

V.1 Topological Manifolds 175
|Ki| = | •
τ i n| � Int|σ n−1|
(remember that •
τi n is the boundary of τi n – see definition in Sect. II.2).
We note that for each p ∈ Int|σ n−1|, the space S(p) is written as the union
S(p) = �r i=1 |Ki|; besides, |Ki|∩|Kj| = | •
σn−1| for each pair of distinct exponents
(i, j) with i, j ∈{1,2,...,n}. By applying the Mayer–Vietoris sequence to the pair
(|K1|,|K2|), we obtain
Hn−1(|K1|∪|K2|;Z) ∼ = Hn−2(| •
σn−1|;Z) ∼ = Z
(note that | •
σn−1| ∼ = Sn−2 ); we next apply the Mayer–Vietoris sequence to the pair
(|K1|∪|K2|,|K3|) and so on, to get the free group with r − 1 generators
� �
r�
Hn−1
|Ki|;Z
i=1
however,
� �
r�
Hn−1 |Ki|;Z ∼= Hn−1(S(p);Z)
i=1
∼ = Z
and so, r − 1 = 1, that is to say, r = 2.
We now assume that |K| is connected. Then by Lemma (II.4.4), K is connected,
in the meaning of the definition given in Sect. II.4 (from the topological point of
view, |K| is connected if and only if it is path-connected).
Let σn ∈ Φ be any n-simplex. We view the simplicial complex K as the union
of two subcomplexes: (a) the simplicial complex L defined by all n-simplexes of K
(and their subsimplexes) that may be linked to σn by a sequence of n-simplexes as
mentioned in 3; (b) the simplicial complex M of all n-simplexes of K for which this
condition does not hold. Note that K = L ∪ M. If M = /0, the assertion is proved;
this occurs when dim|K| = 1 because K is connected; incidentally note that since
K is connected L ∩ M �= /0. We now consider M nonempty and dim|K|≥2. Then,
dim(L ∩ M) ≤ n − 2; in fact, if σn−1 were an (n − 1)-simplex of both L and M,
σn−1 would be a face of both an n-simplex of L and an n-simplex of M, whichis
impossible because of the definitions. For any vertex x of L ∩ M,
dim(S(x) ∩ L)=dim(S(x) ∩ M)=n − 1
and dim(S(x) ∩ L ∩ M) ≤ n − 3. However, because of 2., every(n − 2)-simplex
of S(x) is a face shared by two (n − 1)-simplexes; thus if (S(x) ∩ L)n−1 and
(S(x) ∩ M)n−1 are the sets of all (n − 1)-simplexes of S(x) ∩ L and S(x) ∩ M,
respectively, the chains
cL = ∑
τ∈(S(x)∩L) n−1
τ ,
cM = ∑ τ
τ∈(S(x)∩M)n−1
;