Graphics Reference
In-Depth Information
some additional topics in this section that would be the natural next step for the inter-
ested reader. Since the topics get progressively more advanced, any references for
them will necessarily make for harder and harder reading for someone who has only
learned about algebraic topology from reading this topic. A good general reference is
[Span66].
There are a great many tools for computing homology groups. One of the most
important is the definition of relative homology groups. If L is a subcomplex of a sim-
plicial complex K, then one can define
relative homology groups
H
q
(K,L). These groups
are gotten by looking at the groups C
q
(K)/C
q
(L) and the induced boundary maps
()
()
Æ
()
()
CK
CL
CK
CL
q
q
-
1
∂
r
:
.
q
q
-
1
Then
r
ker
∂
(
)
=
HKL
,
.
q
r
im
∂
q
+
1
(By making the natural definition C
q
(f) = 0, one identifies H
q
(K) with H
q
(K,f).) One
can show that, for q > 0, H
q
(K,L) is isomorphic to H
q
(M), where M is a simplicial
complex that triangulates the quotient space ΩKΩ/ΩLΩ.
Definition.
A sequence of abelian groups and homomorphisms
h
h
h
h
q
+
2
q
+
1
q
q
-
1
...
æÆ
æææÆ
G
ææææææÆ
G
G
ææ
...
q
+
1
q
q
-
1
is said to be an
exact sequence
if ker h
q
= im h
q+1
for all q.
There is an exact sequence
()
Æ
()
Æ
(
)
Æ
()
Æ
()
Æ
...
Æ
HLHKHKLHLHK
q
,
...
q
q
q
-
1
q
-
1
called the
homology sequence of the pair
(K,L) that relates the three homology groups
H
q
(L), H
q
(K), and H
q
(K,L), so that if one knows two of the groups, then the third is
fairly well determined. This is extremely useful in determining the homology groups
of a space from knowledge of the homology groups of subspaces. Simplicial maps on
pairs of complexes induce maps on relative homology groups.
One of the problems with simplicial homology theory is that, although one can
eventually show that it is a topological invariant, this is not obvious at the start since
the groups for a space seem to depend on a particular simplicial subdivision. It would
be nicer if one could define groups that are intrinsically topological invariants. Sin-
gular homology theory provides the answer. In this theory maps of simplices replace
the simplices themselves.
n
Definition.
The simplex D
=
e
0
e
1
...
e
n
is called the
standard n-simplex
.
q
Æ
X
is called a
singular q-simplex
of
X
. Let S
q
be the set of singular simplices of
X
. Define the
group
Definition.
Let
X
be a topological space. A continuous map T :D
