diagram The manner in which this boundary is labeled determines how the space is connected, and therefore the homology of the space.
It is clear that any simple closed curve drawn on the disk for the sphere is a boundary. Therefore, its homology is trivial in dimension 1. The torus has two
classes of nonbounding cycles. When we glue the edges marked “a”, edge “b” becomes a nonbounding 1-cycle and forms a class with all 1-cycles that are
homologous to it. We get a different class of cycles when we glue the edges marked “b.” Each class has a generator, and each generator is free to generate
as many different classes of homologous 1-cycles as it pleases. Therefore, the homology of a torus in dimension 1 is Z
× Z and β
1
= 2. There is a 1-boundary in the diagram, however: the boundary of the disk
that we are gluing. Going around this 1-boundary, we get the description aba
−
1
b
−
1
. That is, the disk makes the cycle with this description a bound- ary. Equivalently, the disk adds the relation aba
−
1
b
−
1
= 1 to the presentation of the group. But this relation is simply stating that the group is Abelian, and
we already knew that. Continuing in this manner, we look at the boundary in the diagram for the
projective plane. Going around, we get the description abab. If we let c = ab,
the boundary is c
2
and we get the definition of the cross-cap used in Conway’s ZIP. The disk adds the relation c
2
= 1 to the group presentation. In other words, we have a cycle c in our manifold that is nonbounding but becomes bounding
when we go around it twice. If we try to generate all the different cycles from this cycle, we just get two classes: the class of cycles homologous to c and
the class of boundaries. But any group with two elements is isomorphic to Z
2
, hence the description of
H
1
. You should convince yourself of the verity of the description of
H
1
for the Klein bottle in a similar fashion.
4.2.4 Invariance
Like the Euler characteristic before it, we defined homology using simplicial complexes. From the definition, it seems that homology is capturing extrin-
sic properties of our representation of a space. We are interested in intrinsic properties of the space, however. We hope that any two different simplicial
complexes K and L with homeomorphic underlying spaces
|K| ≈ |L| have the same homology, the homology of the space itself. Poincaré stated this hope in
terms of “the principal conjecture” in 1904.
Conjecture 4.1 Hauptvermutung Any two triangulations of a topological space have a common refinement.
In other words, the two triangulations can be subdivided until they are the same. This conjecture, like Fermat’s last lemma, is deceptively simple. Pa-
pakyriakopoulos 1943 verified the conjecture for polyhedra of dimension ≤ 2
and Moïse 1953 proved it for three-dimensional manifolds. Unfortunately, the conjecture is false in higher dimensions for general spaces. Milnor 1961
obtained a counterexample for dimensions 6 and greater using Lens spaces. Kirby and Siebenmann 1969 produced manifold counterexamples in 1969.
The conjecture fails to show the invariance of homology Ranicki, 1997.
To settle the question of topological invariance of homology, a more gen- eral theory was introduced, that of singular homology. This theory is defined
using maps on general spaces, thereby eliminating the question of representa- tion. Homology is axiomatized as a sequence of functors with specific prop-
erties. Much of the technical machinery required is for proving that singular homology satisfies the axioms of a homology theory, and that simplicial ho-
mology is equivalent to singular homology. Mathematically speaking, this ma- chinery makes homology less transparent than the fundamental group. Algo-
rithmically, however, simplicial homology is the ideal mechanism to compute topology.
4.2.5 The Euler-Poincaré Formula
To end this section, we derive the invariance of the Euler characteristic Def- inition 4.2 from the invariance of homology. The machinery of homology is
intrinsically beautiful by itself. To catch a glimpse of this beauty, we scruti- nize this relationship with a bit more algebra than we might otherwise need.
Recall that a simplicial complex K gives us a chain complex of finite length. We denote it by
C
∗
. We may now define the Euler characteristic of a chain complex.
Definition 4.15 Euler characteristic of chain complex
χ
C
∗
=
∑
i
−1
i
rank
C
i
. This definition is trivially equivalent to Definition 4.2 as k-simplices are the
generators of
C
k
, or rank
C
i
= s
i
in that definition. So, χK = χ
C
∗
K. If
C
i
is finitely Abelian and not free, we mean by rank the rank of the free part of the group, or its Betti number. We now denote the sequence of homology
functors as
H
∗
Hatcher, 2001. Then,
H
∗
C
∗
is another chain complex: −→
H
n
−→
H
n
−
1
−→ . . . −→
H
1
−→
H
−→ 0. 4.7
ϕ
3
ϕ
1
ϕ
1
ker ϕ
2
im im ϕ
1
ϕ
2
ϕ A
B C
=
Fig. 4.11. Groups in Lemma 4.1. ϕ
2
is injective and ϕ
1
is surjective.
The operators between the homology groups are induced by the boundary op- erators: We map a homology class to the class of the boundary of one of its
members. The Euler characteristic of
H
∗
C
∗
, according to the new definition, is simply ∑
i
−1
i
rank
H
i
= ∑
i
−1
i
β
i
. Surprisingly, the homology functor preserves the Euler characteristic of a chain complex.
Theorem 4.5 Euler-Poincaré χ
C
∗
= χ H
∗
C
∗
. The theorem states that ∑
i
−1
i
s
i
= ∑
i
−1
i
β
i
for a simplicial complex K, deriving the invariance of the Euler characteristic from the invariance of ho-
mology. To prove the theorem, we need a lemma.
Lemma 4.1 Let A , B,C be finitely generated Abelian groups related by the
sequence of maps ϕ
i
:
ϕ
3
−→ A
ϕ
2
−→ B
ϕ
1
−→ C
ϕ
−→ 0, 4.8
where im
ϕ
i
= ker ϕ
i
−
1
. Then, rank B
= rank A + rankC. Proof
The sequence is shown in Figure 4.11. First, we establish two facts. a
ϕ
1
is surjective: im ϕ
1
= ker ϕ = C.
b ϕ
2
is injective: ker ϕ
2
= im ϕ
3
= {e}, so by Corollary 3.1, ϕ
2
is 1-1. By the fundamental homomorphism theorem Theorem 3.10,
B ker ϕ
1
∼ =
im ϕ
1
. By fact a, B ker ϕ
1
∼ = C. Corollary 3.3 gives rank B ker ϕ
1
= rank B
− rank ker ϕ
1
, so rankC = rank B − rank ker ϕ
1
. By fact b, A ∼ =
im ϕ
2
and rank A = rank im ϕ
2
. But im ϕ
2
= ker ϕ
1
, so rank A = rank ker ϕ
1
. Substituting, we get the desired result.
The sequence in the lemma has a name.
Definition 4.16 short exact sequence The sequence in Lemma 4.1 is a short exact sequence
.
