Graphics Reference
In-Depth Information
{
()
}
A
L
)
=
m
SSA
Œ
.
(
,
m
L
Let (j
(L,m)
,K
(L,m)
) denote any geometric realization of
A
(L,m)
and let
X
(L,m)
=ÔK
(L,m)
Ô. Let
c
:
L
Æ
K
(
L
,
m
)
(
L
,
m
)
be the simplicial map defined on a vertex
v
of L by
()
=
(
()
)
c
L
v
jm
v
(
,
m
)
(
L
,
)
and let
p
=
c
:L
Æ
X
.
(
L
,
m
)
(
L
,
m
)
(
L
,
m
)
Definition.
A
(L,m)
is called the abstract simplicial complex
induced
by the labeled
complex (L,m). K
(L,m)
is called a simplicial complex
defined
by (L,m). The space
X
(L,m)
is called a
geometric realization
of (L,m) and the map p
(L,m)
is called the
natural pro-
jection
of ÔLÔ onto
X
(L,m)
.
The labeled complex (L,m) defines the simplicial complex K
(L,m)
and the space
X
(L,m)
uniquely up to isomorphism and homeomorphism, respectively, with
X
(L,m)
just being
a quotient space of ÔLÔ. It is easy to show that c
(L,m)
and p
(L,m)
are an isomorphism
and homeomorphism, respectively, if and only if m is a bijection. A good exercise for
the reader is to return to Figures 6.13(a)-(c) and show that the spaces
X
(L,m)
are in
fact the ones indicated by working through the definitions we have just given.
Although converting labeled figures to the spaces they represent is very easy once one
understands what is going on, one does have to exercise a little caution. For example,
a quick glance at the labeled figure in Figure 6.14(a) might lead one to believe that
one is representing a cylinder. This is incorrect. The space
X
(L,m)
is actually homeo-
morphic to the sphere
S
2
, just like Figure 6.13(c). There is also a danger in using too
(L,m)
(L,m)
v
0
v
0
v
2
v
0
v
0
v
0
v
1
v
3
v
1
K
(L,m)
≈∂
s, where
is a 3-simplex
X
(L,m)
≈ S
2
σ
X
(L,m
)
≈ v
0
(a)
(b)
Figure 6.14.
Why labeled figures have to be interpreted carefully.
