Graphics Reference
In-Depth Information
Let us show how homology groups can be used to tackle the problem of classify-
ing topological spaces. At the moment homology groups have only been defined for
simplicial complexes, but we could define the homology groups of a polyhedron to be
the homology groups of a simplicial complex that triangulates it. The problem is that
spaces can be triangulated in many different ways. Therefore, the next theorem, which
asserts the topological invariance of homology groups, is essential here. Its proof will
be given shortly.
7.2.1.9. Theorem.
Suppose that K and L are simplicial complexes and that ΩKΩ is
homeomorphic to ΩLΩ. Then H
q
(K) is isomorphic to H
q
(L) for all q.
Proof.
This is an immediate corollary of Theorem 7.2.3.1 in Section 7.2.3 since
homeomorphisms are homotopy equivalences.
Definition.
Let
X
be a polyhedron. Choose any triangulation (K,j) for
X
and define
the
qth homology group
of
X
, H
q
(
X
), by H
q
(
X
) = H
q
(K). The
qth Betti number
of
X
,
b
q
(
X
), and the
qth torsion coefficients
of
X
are defined to be the rank and torsion coef-
ficients of K, respectively.
Although the groups H
q
(
X
) are not uniquely defined since a polyhedron has many
triangulations, they are well defined up to isomorphism by Theorem 7.2.1.9. This is
all we need to be able to conclude that homology groups give us topological invari-
ants of spaces and not just properties of particular triangulations and are therefore
exactly the type of computable algebraic invariants we were looking for in Section
6.2. See [AgoM76] for a wide range of applications justifying the hard work we put
into our effort. Some of these will also be discussed in Section 7.2.3. The next chapter
will also look into the subject further, but at this point we are lacking one important
ingredient of the theory, namely, we do not yet know how homology groups behave
with respect to maps.
Table 7.2.1.1 summarizes the computations we have made so far and some that
come from Exercises 7.2.1.3-4. The results are stated for polyhedra rather than sim-
plicial complexes because that is what we are after anyway, not the intermediate struc-
tures, the simplicial complexes.
We finish this section with a comment about the concept of orientability. Looking
at Table 7.2.1.1 we see that what distinguishes closed compact orientable surfaces
S
from nonorientable ones is the group H
2
(
S
), or, to put it another way, the ability to
find a nonzero 2-cycle. A little reflection shows that we got a nonzero 2-cycle precisely
because it was possible to orient the 2-simplices of a triangulating simplicial complex
so that adjacent 2-simplices induced the opposite orientation on the 1-simplex in their
boundary which they had in common. This shows that the condition of orientability
as described in Chapter 1 is actually easily computed by combinatorial methods - one
simply needs to find a 2-cycle.
7.2.2
Induced Maps
In Section 7.2.1 we defined a mapping from simplicial complexes K to their homol-
ogy groups H
q
(K). We now show how this mapping of
objects
extends to a mapping
of
maps
in a natural way.
