Graphics Reference
In-Depth Information
s
s
()
Æ
s
()
f
:
C
X
C
Y
#
q
by the condition that f
#q
(T) = f
°
T for every singular q-simplex T :D
q
Æ
X
.
One can show that ∂
s
°
f
#q
= f
#q-1
°
∂
s
, so that the maps f
#q
induce well-defined
homomorphisms
s
s
()
Æ
s
()
f
:
H
X
H
Y
.
*
With the groups H
s
(
X
), their corresponding relative groups H
s
(
X
,
A
) for a subspace
A
of
X
, and maps f
*q
, along with their relative analogs, we never have to worry about
triangulations. The topological invariance is trivially built into the definition. Fur-
thermore, polyhedra now have real groups associated to them, not groups up to iso-
morphism. The nontrivial part is showing that they give the same groups as the
simplicial homology theory. The solution to this problem comes from the fact, referred
to earlier in Section 7.2.3, that homology theories can be axiomatized using the
Eilenberg-Steenrod axioms. We can take our various definitions of homology groups
simply to be existence results that assert that there are objects that satisfy the abstract
theory. Any two theories that satisfy the axioms will have isomorphic groups if they
have isomorphic homology groups for a point.
Since cohomology groups are derived algebraically from chain groups, one can
obviously define
singular cohomology groups
.
Although homotopy groups are much harder to compute than homology groups,
there are tools that help in this. One such is the fact that one can define relative homo-
topy groups p
n
(
X
,
A
,
x
0
) that play the same role for homotopy theory that the relative
homology groups play for homology theory. Given a topological space
X
, a subspace
A
,
and a point
x
0
Œ
A
, these groups are obtained from relative homotopy classes of maps
(
)
Æ
(
nn
1
)
a
:
IIe
,
∂
,
XAx
,
,
,
0
where the homotopies have to keep mapping ∂
I
n
to
A
. Maps between pairs of spaces
induce homomorphisms of the relative groups. There is also an exact sequence
(
)
Æ
(
)
Æ
(
)
Æ
(
)
Æ
...
Æ
p
X A x
, ,
p
A x
,
p
X x
,
p
X A x
, ,
....
n
+
1
0
n
0
n
0
n
0
From an abstract point of view, we can think of H
*
and p
*
as examples of “func-
tors” from the “category” of topological spaces to the “category” of groups. (The reason
for the quotes around some terms is that they have precise mathematical definitions
that we cannot go into here.) This is how topological questions get translated into
algebraic questions.
The theories we have talked about, homology, homotopy, and so on, really apply
to arbitrary topological spaces. Of course manifolds are the most interesting ones, in
particular three-dimensional manifolds, because those are the spaces with which we
have contact in everyday life. Therefore, it should not be surprising that a great deal
of work has been done in low-dimensional topology. Unfortunately, we shall see in the
next chapter that, as counter-intuitive as it might seem, a lot more is known about n-
dimensional manifolds for n ≥ 5 than three- and four-dimensional manifolds. As a
starting point for more information on advanced aspects of this subject we suggest
the topics [Mois77] and [Matv03].
