Graphics Reference
In-Depth Information
Definition.
An
abstract simplicial complex
is a set
A
of nonempty subsets of a given
set V such that
(1) {
v
} Œ
A
for every
v
ΠV, and
(2) if S Œ
A
, then every nonempty subset of S belongs to
A
.
The elements of
A
are called
(abstract) simplices
. If S Œ
A
and if S has k + 1 elements,
then S is called an
(abstract) k
-
simplex
. The elements of V are called the
vertices
of
A
and one identifies the vertex
v
in V with the 0-simplex {
v
} in
A
.
Every simplicial complex K defines an abstract simplicial complex
A
K
in a natural
way, namely,
{
{
}
}
A
K
=
vv
,,,
K
v
vv v
L
k
is a k simplex of K
-
.
01
k
01
It is easy to check that
A
K
is in fact an abstract simplicial complex. For example, if L
is as in Figure 6.10, then
{
{
} {
}
}
A
L
=
vvv vv
,,, , , ,
vv
.
012
02
12
Conversely, it is possible to associate a geometric complex to an abstract simplicial
complex.
Definition.
Let
A
be an abstract simplicial complex. A
geometric realization
of
A
is
a pair (j,K), where K is a simplicial complex and j is a bijective map from the ver-
tices of
A
to the vertices of K such that {
v
0
,
v
1
,...,
v
k
} is a k-simplex of
A
if and only
if j(
v
0
)j(
v
1
). . .j(
v
k
) is a k-simplex of K.
To show that geometric realizations exist, we simply need to “fill in” the missing
points in the abstract simplices.
6.4.1. Theorem.
Every abstract simplicial complex
A
has a unique (up to isomor-
phism) geometric realization.
Proof.
Let V be the set of vertices of
A
and assume that V has n + 1 points. Let j
be a bijection between V and the set of vertices of any n-simplex s in
R
n
. Define a
subcomplex K of s by
{
()() ()
{
}
}
K
=
jj
vv
L
j
v vv v
,,,
K
is a k simplex of
-
A
.
0
1
k
0
1
k
It is easy to see that (j,K) is a geometric realization of
A
. If (j¢,K¢) is another geo-
metric realization of
A
, then j
°
j
-1
: K Æ K¢ is an isomorphism.
Theorem 6.4.1 is much more significant than one might conclude from its trivial
proof. It is this theorem that allows us to define certain quotient spaces without any
fancy point set topology.
Before we show how abstract complexes can be used to make sense of the cutting
and pasting operations that we referred to at the beginning of this section, it will help
the reader understand what we are talking about here by giving some examples. Prob-
