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to hold. If we treat simplices as formal symbols, then one way to satisfy this equation
is to identify the expression t
3
+ t
3
(which one is also tempted to write as 2t
3
) with 0.
A homology theory based on this approach (the mod 2 homology groups) will be dis-
cussed in Section 7.2.6. In this section we describe a second approach. It may seem
slightly more complicated initially but will lead to better invariants.
First of all, we introduce a notion of orientation for simplices. For now, we shall
leave things to intuition and infer the orientation of a simplex simply from the order
in which its vertices are listed. For example, the expression
vw
will be used to deter-
mine not only the 1-simplex with vertices
v
and
w
but also the direction (in this case
from
v
to
w
) in which one should “walk” if one were to walk along that edge. Given
an orientation of a 2-simplex, we get a natural orientation of the boundary curve. For
example, returning to the simplicial complex in Figure 7.2, using the expression
v
0
v
1
v
2
for the 2-simplex s
1
indicates that it has been oriented in the counterclockwise fashion
and its boundary will be thought of as a closed path that is intended to be traversed
also in a counterclockwise fashion, say by starting at
v
0
and then walking from
v
0
to
v
1
, from
v
1
to
v
2
, and finally from
v
2
back to
v
0
.
We can express the relationships between oriented simplices and their boundaries
symbolically by introducing a boundary operator ∂*, so that what we were just saying
can be summarized by equations of the form
∂
*
(
)
=
vvv
vv
+
vv
+
vv
012
01
12
20
and
∂
*
(
)
=
vvv
vv
+
vv
+
v v
21
.
132
13
32
More generally, since arbitrary oriented regions also define an orientation on their
boundary, it makes sense to have the operator ∂* defined on those. In the case of the
union of the two simplices s
1
and s
2
the geometry would imply that
∂
*
(
)
=
(7.1)
vvv
»
vvv
vv
+
vv
+
vv
+
vv
20
.
012
132
01
13
32
On the other hand, from an algebraic point of view we would like
*
*
(
*
(
(
)
=
)
+
)
∂
vvv
+
vvv
∂
vvv
∂
vvv
012
132
012
132
(7.2)
(
)
+
(
)
=
vv
+
vv
+
vv
vv
+
vv
+
vv
21
.
01
12
20
13
32
The difference between the two expressions on the right of equations (7.1) and (7.2)
is that the second has an extra
v
1
v
2
+
v
2
v
1
term. In terms of walking along paths, the
difference between the two paths is that in the second we took extra strolls, first from
v
1
to
v
2
and later from
v
2
back to
v
1
. This suggests that we should identify
v
1
v
2
+
v
2
v
1
with 0. By formally defining
v
2
v
1
=-
v
1
v
2
we make this even more plausible. Geomet-
rically, it means that -
v
1
v
2
represents the path from
v
1
to
v
2
traversed in the opposite
direction. We would then have the reasonable looking equations
(
)
=
vv
+
v v
=
vv
+ -
vv
0
.
12
21
12
12
This as far as we go in our motivational discussion and we now start our rigorous devel-
opment of homology groups. What we tried to indicate was that if one is interested in
studying the holes in a space, then one approach to this leads to symbol manipulation
involving oriented simplices, formal linear sums of these, and boundary maps. A
homology group will be a “group of cycles” modulo a “group of boundaries.”
