Graphics Reference
In-Depth Information
is also the zero map. Next, note that the group H
0
(K) is generated by the element
v
0
+ B
0
(K) and
(
()
)
=
()
+
()
=+
()
f
v
+
B
Kf
v
B
K B
v
K
.
*
00
0
#
00
0
0
0
This implies that f
*0
is the identity map.
7.2.2.5. Example.
To compute f
*q
when f is defined by f(
v
0
) =
v
0
, f(
v
1
) =
v
2
, and f(
v
2
)
=
v
1
.
Solution.
It follows from an argument similar to the one in the previous example
that f
*q
= 0 if q > 1 and f
*0
is the identity map. Only f
*1
is different this time. Recall
from Example 7.2.1.5 that
=
[
(
]
+
[
]
+
[
]
)
+
()
a
vv
vv
vv
B
K
01
12
20
1
is a generator of H
1
(K). Since
[
]
+
[
]
+
[
]
*
()
=
(
)
+
()
fa f
vv
vv
vv
BK
1
#
1
01
12
20
1
[
]
+
[
]
+
[
]
(
()()
()( )
()()
)
+
()
=
f
vv
f
f
vv
f
f
vv
f
BK
0
1
1
2
2
0
1
=
[
]
+
[
]
+
[
]
(
)
+
()
vv
vv
vv
BK
02
21
10
1
=-
a
it follows that f
*1
is the negative of the identity map.
The next lemma lists some basic properties of the maps f
#q
and f
*q
that are easy
to prove.
7.2.2.6. Lemma.
Let f : K Æ L and g : L Æ M be simplicial maps between simplicial
complexes. Then
(1) (g
°
f)
#q
= g
#q
°
f
#q
:C
q
(K) Æ C
q
(M).
(2) (g
°
f)
*q
= g
*q
°
f
*q
:H
q
(K) Æ H
q
(M).
(3) If K = L and f = 1
K
, then f
#q
and f
*q
are also the identity homomorphisms.
Proof.
This is Exercise 7.2.2.1.
Now simplicial complexes and maps are basically only tools for studying topo-
logical spaces and continuous maps. We shall show next how continuous maps induce
homomorphisms on homology groups.
Definition.
Let K and L be simplicial complexes and suppose that f : ΩKΩÆΩLΩ is a
continuous map. A
simplicial approximation to f
is a simplicial map j :K Æ L with the
following property: If
x
ŒΩKΩ and if f(
x
) Œ s for some simplex s Œ L, then ΩjΩ (
x
) Πs.
The next lemma summarizes two important properties of simplicial
approximations.
