Simplicial Structures in Topology

174 V Triangulable Manifolds
other at an n − 1 face or they are linked by a chain of n-simplexes. The proof of
these results is based on an argument found in the next lemma; we ask the reader
to remember that we may associate the closed subspace S(p), thatistosay,the
boundary of the space
D(p)= �
|σ|,
σ∈B(p)
with each p ∈|K|, whereB(p) is the set of all σ ∈ Φ such that p ∈|σ| (see
Sect. II.2).
(V.1.4) Lemma. Let X ∼ = |K| be a triangulable n-manifold; then for whatever p ∈
|K|,S(p) is of the same homotopy type as the sphere S n−1 .
Proof. Let (U,φ) be a chart of |K| containing the point p. Sinceφ(U) is homeomorphic
to an open set W ⊂ Rn , there is a closed disk Dn ε (φ(p)), with center φ(p)
and radius ε, contained in W . The restriction of φ −1 to Dn ε(φ(p)) is a homeomorphism
from Dn ε (φ(p)) into a subspace of |K| containing p. We recall that Dnε (φ(p))
is a triangulable space; let L be a simplicial complex whose geometric realization is
identified to Dn ε(φ(p)). We now consider the homeomorphism φ −1 : |L|→|K| and
apply Theorem (II.2.11) to conclude that S(φ(p)) ∼ S(p). We complete the proof
by noting that S(φ(p)) ∼ = Sn−1 . �
The following result is very important (cf. [24, Theorem 5.3.3]).
(V.1.5) Theorem. Any triangulable n-manifold |K| has the following properties:
1. dimK = n
2. For every vertex p ∈|K|, there is an n-simplex σ of K such that p ∈|σ|
3. Every (n − 1)-simplex of K is a face of exactly two n-simplexes
Moreover, if |K| is connected, for any two n-simplexes σ and τ of K, there is a
sequence of n-simplexes σ = σ1,...,σr = τ such that σi ∩σi+1 is an (n−1)-simplex
for every i = 1,...,r − 1.
Proof. 1. Lemma (V.1.4) shows that Hn−1(S(p);Z) ∼ = Z for each p ∈|K|. Hence,
if dimK < n, foreveryp∈|K|, dimS(p) < n − 1; therefore, Hn−1(S(p);Z) =0,
contradicting the lemma. We now suppose that dimK = m > n; this means that
the simplicial complex K has at least one m-simplex with m > n and so, for every
point p in the interior of |σ|, the space S(p) is of the same homotopy type as Sm−1 ,
once again in contradiction to Lemma (V.1.4) (two spheres of different dimensions
cannot be of the same homotopy type).
2. If K had no n-simplex with geometric realization containing p, then the dimension
of the simplicial complex S(p) would be strictly less than n − 1 and thus S(p)
could not be of the same homotopy type as Sn−1 .
3. Let us suppose σn−1 to be a face of rn-dimensional simplexes τi n , i = 1,...,r. Let
Int|σn−1| = |σ n−1| � | •
σn−1|
be the interior of |σn−1|; foreveryi = 1,...,r, wedefine