Simplicial Structures in Topology

48 II Simplicial Complexes
II.2.1 The Geometric Realization Functor
For a given simplicial complex K =(X,Φ), letV(K) be the set of all functions
p: X → R≥0 (nonnegative real numbers); we define the support of an arbitrary
p ∈ V(K) to be the finite set
Let |K| be the set defined as follows:
We now define the function
s(p)={x ∈ X | p(x) > 0}.
|K| = {p ∈ V(K) | s(p) ∈ Φ and ∑
x∈s(p)
p(x)=1}.
d : |K|×|K|−→R≥0
which takes any pair (p,q) ∈|K|×|K| into the real number
�
d(p,q)=
∑
x∈X
(p(x) − q(x)) 2 .
This function is a metric on K (verify the conditions defining a metric given in
Sect. I.1.5); hence, it defines a (metric) topology on |K|. The metric space |K| is the
geometric realization of K. Observe that |K| is a bounded space, in the sense that
(∀p,q ∈|K|), d(p,q) ≤ √ 2. Moreover, |K| is a Hausdorff space.
We can write the elements of K as finite linear combinations. In fact, for each
vertex x of K, with a slight abuse of language, let us denote with x the function
of V(K), with value 1 at the vertex x and 0 at any other vertex; in a more formal
fashion,
�
0ify�= x
(∀y ∈ X) x(y)=
1ify = x
(in other words, we identify the vertex x with the corresponding real function of
V(K), whose support coincides with the set {x}). Hence if s(p)={x0,x1,...,xn} is
the support of p ∈|K|, and assuming that p(xi)=αi, i = 0,1,...,n, we can write p as
p =
n
∑
i=0
αixi.
The real numbers αi , i = 0,...,n,arethebarycentric coordinates of p (in agreement
with the barycentric coordinates defined by n+1 independent points of an Euclidean
space).
(II.2.6) Remark. Because K has a finite number of vertices, say n, we can embed
the set of vertices X in the Euclidean space Rn , so that the images of the elements of
X coincide with the vectors of the standard basis. Then we can take the convex hulls
in Rn of the vectors corresponding to the simplexes of K, to obtain an Euclidean