Graphics Reference
In-Depth Information
inductively by
n
sd
=
zero map for n
,
<
0
,
#
0
()
sd
=
identity map of C
K
,
0
#
q
n
n
-
1
sd
=
sd
o
sd
,
for n
>
0
.
#
q
#
#
The maps sd
#q
induce homomorphisms (actually isomorphisms)
(
)
n
()
Æ
n
()
sd
*
:
H
K
H
sd
K
.
q
q
We are now ready to show how continuous maps induce homomorphisms on
homology groups. Let K and L be simplicial complexes and let
fK L
:
Æ
be a continuous map. The Simplicial Approximation Theorem implies that there is an
n ≥ 0, such that f admits a simplicial approximation
n
()
Æ
j :
sd
K
L
.
Definition.
The homomorphism
()
Æ
()
f
:
HK HL
*
q
q
q
defined by
n
f
=j
o
sd
*
q
*
q
*
is called the
homomorphism induced on the qth homology group
by the continuous
map f.
7.2.2.10. Lemma.
(1) f
*q
is a well-defined homomorphism.
(2) If K = L and f = 1
K
, then f
*q
is the identity homomorphism.
(3) If M is a simplicial complex and g : ΩLΩÆ ΩMΩ is a continuous map, then
(g
°
f)
*q
= g
*q
°
f
*q
.
Proof.
See [AgoM76].
7.2.2.11. Theorem.
Let K and L be simplicial complexes and suppose that f,
g:ΩKΩÆΩLΩ are continuous maps that are homotopic. Then f
*q
= g
*q
:H
q
(K) Æ H
q
(L)
for all q.
Proof.
See [AgoM76].
