Graphics Reference
In-Depth Information
T
b
b
g
C
D
a
a
Figure 7.1.
Holes in a torus.
v
3
t
5
v
2
s
2
t
2
t
3
t
4
K
s
1
v
0
t
1
v
1
Figure 7.2.
Boundaries of simplices in complexes.
we need to look at some sort of equivalence classes of closed curves with respect to
a suitable equivalence relation. One natural such equivalence relation would be homo-
topy. This is what one uses for the definition of homotopy groups. For homology
groups we shall use a weaker and more algebraic notion.
Note that the curve g in
C
bounds a disk
D
and that the union of the two curves
a and b in
C
also bounds a region. This suggests that we define an equivalence rela-
tion for closed curves where the equivalence class that represents the trivial element
corresponds to curves that bound. We continue this line of thought in the context of
simplicial complexes. One of the useful aspects of simplicial complexes is that they
have both a geometric and an abstract nature to them. It is the latter that we want to
take advantage of right now because it will let us switch into a symbol manipulation
mode where purely algebraic manipulations replace geometric operations.
Figure 7.2 shows two 2-simplices, s
1
=
v
0
v
1
v
2
and s
2
=
v
1
v
2
v
3
, in a simplicial
complex K. Consider the edges t
1
=
v
0
v
1
, t
2
=
v
0
v
2
, t
3
=
v
1
v
2
, t
4
=
v
1
v
3
, and t
5
=
v
2
v
3
in
K. As point sets, the boundaries of s
1
, s
2
, and s
1
» s
2
are t
1
» t
2
» t
3
, t
3
» t
4
» t
5
,
and t
1
» t
2
» t
4
» t
5
, respectively. What is the relationship between the boundary of
the region s
1
» s
2
and the boundaries of the individual simplices s
1
and s
2
? We could
say that the edge t
3
, considered as part of the boundary of s
1
, and the edge t
3
, con-
sidered as part of the boundary of s
2
, have, in some sense, cancelled each other. This
idea of cancellation can be made more meaningful if we use a more suggestive alge-
braic notation and write “+” instead of “».” If we want the function “boundary of” to
be additive, then we want the equation
(
)
+++
(
)
tttt ttt ttt
+++ = ++
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