We use the lemma to prove the Euler-Poincaré relation. Proof
[Euler-Poincaré] Consider the following sequences: −→
Z
n i
−→
C
n ∂
n
−→
B
n
−
1
−→ −→
B
n i
−→
Z
n ϕ
−→
H
n
−→ 0, where 0 is the zero map, i is the inclusion map, and
ϕ assigns to a cycle z ∈
Z
n
its homology class [z] ∈
H
n
. Both sequences are short exact. Applying Lemma 4.1, we get:
rank
C
n
= rank
Z
n
+ rank
B
n
−
1
, 4.9
rank
Z
n
= rank
B
n
+ rank
H
n
. 4.10
Substituting the second equation into the first, multiplying by −1
n
, and sum- ming over n gives the theorem.
4.3 Arbitrary Coefficients
We spent a considerable amount of energy in Sections 3.3.3 and 3.3.4 extend- ing the fundamental theorem of finitely generated Abelian groups to arbitrary
R -modules. We now take advantage of our effort to generate additional homol-
ogy groups rather quickly. Recall that any finitely generated group is also a Z- module. In this view, we are multiplying elements of a homology group with
coefficients from the ring of integers. We may replace this ring with any PID
D , such as Z
2
, and the fundamental theorem of finitely generated D-modules Theorem 3.19 would give us a factorization of the homology groups in terms
of the module. This fact generates a large number of homology groups, for which we need new notation.
Definition 4.17 homology with coefficients The
k th
homology group
with ring of coefficients D is
H
k
K; D =
Z
k
K; D
B
k
K; D. If we choose a field F as set of coefficients, the homology groups become
vector spaces with no torsion:
H
k
K; F ∼ = F
r
, where r is the rank of the vector space. A natural question is whether homology groups generated with differ-
ent coefficients are related. The Universal Coefficient Theorem for Homology answers in the affirmative, relating all types of homology to Z homology. Be-
fore stating the theorem, we need to look at two new functors that the theorem uses. I will not define these functors formally, as they are large and very inter-
esting topics by themselves. Rather, I aim here to state the properties of these functors that allow us to understand the theorem and use it for computation.
Table 4.3. Rules for computing tensor and torsion products, given for general Abelian groups G and certain type of groups: Z
m
and F fields.
tensor ⊗
torsion ∗
G Z
⊗ G ∼ = G
Z ∗ G ∼
= {0} G
Z
n
⊗ G ∼ = GnG
Z
n
∗ G ∼ = ker G
n
→ G Z
m
Z ⊗ Z
m
∼ = Z
m
Z ∗ Z
m
∼ = {0}
Z
m
Z
n
⊗ Z
m
∼ = ZdZ, d = gcdn, m
Z
n
∗ Z
m
∼ = ZdZ, d = gcdn, m
F Z
⊗ F ∼ = F
Z ∗ F ∼
= {0} F
Z
n
⊗ F ∼ = {0}
Z
n
∗ F ∼ = {0}
The first functor we need is the tensor product, which maps two Abelian groups to an Abelian group. The tensor product of Abelian groups A and B,
denoted A ⊗ B, is like the product A × B, except that all functions on A ⊗ B are
bilinear. The tensor is commutative, associative, and has distributive properties with respect to group products. The distributive properties are easier to grasp
by thinking of direct products as direct sums, as is often the case when the groups are Abelian. The universal theorem uses the tensor product to rename
the factors of a product.
The other functor we need is the torsion product, which also maps two Abelian groups to an Abelian group. Intuitively, the torsion product of Abelian
groups A , B, denoted A ∗ B, captures the torsion elements of A with respect to
B . The torsion functor is also commutative and has distributive properties. If
either A or B is torsion-free that is, it is free, A ∗ B = 0, the trivial group.
Table 4.3 gives rules for computing using the torsion and tensor products. The rules look cryptic, but they match our intuition of these functors. For example,
note how the tensor product translates between Z and a group G. Along with the distributive properties, we use the tensor product to translate between direct
products representing the structure of homology groups. We are now ready to tackle the universal theorem.
Theorem 4.6 universal coefficient Let G be an Abelian. The following se- quence is short exact:
−→ H
k
K ⊗ G −→ H
k
K; G −→ H
k
−
1
K ∗ G −→ 0. 4.11
Let us use the rules from Table 4.3 to see what the theorem states for the following two cases: homology with coefficients in Z
p
, where p is prime, and
a field F. We know by Theorem 3.10 that
H
k
K ∼ = Z
d
1
× Z
d
2
× · · · × Z
d
n
× Z
β
k
, 4.12
where d
i
is the appropriate prime power and β
k
is the kth Betti number. We would like to know how the ring of coefficients changes this result in
H
k
K; Z
p
and
H
k
K; F. 1. Case
H
k
K; Z
p
: Applying the tensor with Z
p
and distributing over the factors, we get
H
k
K ⊗ Z
p
∼ = Z
d
1
pZ
d
1
× · · · × Z
d
n
pZ
d
n
× Z
p β
k
. 4.13
On the right side of sequence 4.11, the torsion functor eliminates the Z
factors and modifies the torsion coefficients, giving us
H
k
−
1
K ∗ Z
p
∼ = Z
c
1
× Z
c
2
× · · · × Z
c
m
, 4.14
where c
i
are the corresponding gcd’s. In this case, the sequence splits and we get:
H
k
K; Z
p
∼ =
H
k
K ⊗ Z
p
×
H
k
−
1
K ∗ Z
p
4.15 ∼
= Z
d
1
pZ
d
1
× · · · × Z
d
n
pZ
d
n
× 4.16
Z
c
1
× · · · × Z
c
m
× Z
p β
k
. Therefore, by using Z
p
as the ring of coefficients, we get the same Betti numbers as before, but different torsion coefficients.
2. Case
H
k
K; F: According to the rules,
H
k
−
1
K ∗ F ∼ = {0}, reducing
sequence 4.11 to −→
H
k
K ⊗ F
ϕ
−→
H
k
K; F. −→ 0 4.17
Applying the facts in the proof of Lemma 4.1 shows that ϕ is both
injective and surjective. In other words,
H
k
K ⊗ F ∼ =
H
k
K; F. The tensor product eliminates the torsion factors from
H
k
and renames the Z
factors, so
H
k
K; F ∼ =
H
k
K ⊗ F ∼ = F
β
k
. We lose the torsion and get the same Betti numbers whenever we use a field of coefficients for
computation. We restate our results in a corollary.
Corollary 4.2 Let p be a prime and F be a field. Then,
H
k
K; Z
p
∼ =
H
k
K ⊗ Z
p
× H
k
−
1
K ∗ Z
p
, 4.18
H
k
K; F ∼ = F
β
k
. 4.19
