Graphics Reference
In-Depth Information
Table 7.2.1.1
Some spaces and their homology groups.
X
H
0
(
X
)
H
1
(
X
)
H
2
(
X
)
H
i
(
X
)
i
>
2
D
0
Z
0
0
0
S
0
Z
≈
Z
0
0
0
S
1
Z
Z
0
0
S
2
Z
0
Z
0
S
1
¥
S
1
Z
Z
≈
Z
Z
0
P
2
Z
Z
2
0
0
The Klein bottle
Z
Z
≈
Z
2
0
0
Orientable surface (genus k)
Z
k (
Z
≈
Z
)
Z
0
Z
k
-
1
Nonorientable surface (genus k)
Z
≈
Z
2
0
0
Definition.
Let K and L be simplicial complexes and let f : K Æ L be a simplicial
map. Define maps
()
Æ
()
f
#
:
CKCL
q
q
q
as follows:
If q < 0 or q > dim K, then f
#q
= 0.
If 0 £ q £ dim K, then f
#q
is the unique homomorphism defined by the condition
that
(
[
]
)
=
[
()()
◊◊◊
()
]
()
π
()
f
vv
◊◊◊
v
f
v
f
v
f
v
,
if f
v
f
v
for i
π
j
,
#
q
01
q
0
1
q
i
j
=
0
,
otherwise
,
for each oriented q-simplex [
v
0
v
1
···
v
q
] of C
q
(K). (The map f
#q
is well defined
because the group C
q
(K) is a free group with generators [
v
0
v
1
···
v
q
].)
7.2.2.1. Lemma.
∂
q
°
f
#q
= f
#q-1
°
∂
q
for all q. In other words, for all q there is a com-
mutative diagram
f
()
æÆ
#
q
()
CK
æ
CL
q
q
∂
Ø Ø
()
æÆ
∂
q
q
()
CK
ææ
CL
.
q
-
1
q
-
1
f
#
q
-
1
Proof.
Clearly, it suffices to show that
)
=
(
)
(
)
(
[
]
(
[
]
)
∂
o
f
vv
◊◊◊
v
f
o
∂
vv
◊◊◊
v
q
#
q
01
q
#
q
01
q
q
-
1
for every oriented q-simplex [
v
0
v
1
···
v
q
] in K and we do this by computing both sides
of this equation.
Case 1.
The vertices f(
v
0
), f(
v
1
),..., and f(
v
q
) are all distinct.
