Download SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR

SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR
HOMOLOGY
Given a space X and an abelian group A, our next goal is to express singular homology H∗ (X; A)
in terms of the integral homology H∗ (X) ∼
= H∗ (X; Z) and the coefficient group A. A naive guess
would be that forming the tensor product and forming homology would commute, i.e., that we
simply have a formula
∼ Hk (X) ⊗ A, k ≥ 0,
Hk (X; A) =
but this turns out to be an unreasonable demand. However, there is an universal coefficient theorem
relating homology with coefficients and integral homology. This theorem is a reminiscence of the
fact that, in general, the tensor product is right exact but not necessarily left exact.
Recall that the formation of homology is defined as the passage to subquotients (‘cycles modulo
boundary’). Thus, if we wanted the above formula to be true we should better assume that the
functor (−) ⊗ A preserves subobjects and quotient objects. This is fine for quotients but, in general,
subobject are not preserved.
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Lemma 1. Let B be an abelian group and let 0 → A0 → A → A00 → 0 be a short exact sequence of
abelian groups. Then A0 ⊗ B → A ⊗ B → A00 ⊗ B → 0 is exact.
Proof. This can be proved by abstract nonsense. We know that (−) ⊗ B is a left adjoint functor
(Exercise 6 on Exercise Sheet 1) and hence preserves colimits and, in particular, cokernels. In the
exercises you are asked to give a proof in more elementary terms.
The statement of the lemma is referred to by saying that the functor (−) ⊗ B is right exact.
There is the dual notion of a left exact functor Ab → Ab and the combined notion of an exact
functor, but these properties are, in general, not enjoyed by the tensor product.
Example 2. We give two examples showing that the tensor product, in general, is not left exact
and one example of an exact tensor product functor.
(1) Let m ∈ Z be different from −1, 0, 1 and let us consider the short exact sequence
m
0 → Z → Z → Z/mZ → 0.
If we apply − ⊗ B to it, then we obtain the exact sequence
Z ⊗ B → Z ⊗ B → Z/mZ ⊗ B → 0.
m
Up to natural isomorphism it can be rewritten as B → B → Z/mZ ⊗ B → 0 and hence
yields an isomorphism Z/mZ ⊗ B ∼
= B/mB. Note that m : B → B is injective if and only if
B has no m-torsion elements. In particular, if we specialize to B = Z/mZ then we obtain
the exact sequence
m
Z/mZ → Z/mZ → Z/mZ ⊗ Z/mZ → 0,
and the first map is clearly not injective as it is trivial.
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SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
(2) In this example we use fact that for torsion groups A we have an isomorphism Q ⊗ A ∼
=0
(Exercise!). Given such a torsion group, let us form its tensor product with the short exact
sequence
0 → Z → Q → Q/Z → 0
in order to obtain the exact sequence
Z ⊗ A → Q ⊗ A → Q/Z ⊗ A → 0.
But for non-trivial torsion groups the first map is certainly not injective since it is isomorphic
to the map A → 0.
(3) Let F be a free abelian group and let 0 → A0 → A → A00 → 0 be exact. Then also
0 → A0 ⊗ F → A ⊗ F → A00 ⊗ F → 0
is a short exact sequence. This is immediate if F is free of rank one. In the general case, it
follows from the additivity of the tensor product together with the fact that a direct sum
of short exact sequences is again a short exact sequence.
We would now like to measure in a systematic way the deviation of the tensor product from being
left exact. This is done by means of torsion products of abelian groups and we include a short detour
discussing them. We begin with the following seemingly ad-hoc construction. The justification for
this construction is given by Theorem 11. For now just observe that in the construction we ‘resolve’
arbitrary abelian groups by free ones which we know from the previous example to be well-behaved
with respect to the formation of tensor products.
Construction 3. Let A be an abelian group. A free presentation of A is a short exact sequence
of abelian groups
0 → F1 → F0 → A → 0
such that F0 and F1 are free. Invoking a standard fact from algebra, it is easy toL
see that such free
presentations always exist. For example, we could consider the free abelian group a∈A
L Z generated
by all elements of A. This free abelian group comes with an obvious surjection
a∈A Z → A
mapping the generator ea to a. Thus, we obtain an associated short exact sequence
M
0→K→
Z → A → 0,
a∈A
and, using the algebraic fact that subgroups of free abelian groups are again free, we can conclude
that this defines a free presentation of A.
Given an arbitrary free presentation of A, we take this short exact sequence and read it as a
morphism of chain complexes
···
/0
/0
/ F1
/ F0
···
/0
/0
/0
/ A.
Note that the fact that we started with a short exact sequence precisely means that this chain map
F → A is a quasi-isomorphism. Here we commited abuse of notation and simply wrote A for the
above chain complex concentrated in degree zero, while F = (. . . → 0 → F1 → F0 ) is concentrated
in dimensions 0 and 1.
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
3
Let now B be a further abelian group and let F ⊗ B ∈ Ch≥0 (Z) be the levelwise tensor product,
still concentrated in dimensions 0, 1. Then we make the following definitions,
Tork,F (A, B) := Hk (F ⊗ B), k = 0, 1.
∼ A ⊗ B. In fact, Tor0,F (A, B) is defined
Let us observe that we have an isomorphism Tor0,F (A, B) =
as H0 (F ⊗ B), i.e., as the 0-th homology of (. . . → 0 → F1 ⊗ B → F0 ⊗ B). But this is just a
fancy way of saying that we consider the cokernel of F1 ⊗ B → F0 ⊗ B. Using the right-exactness
of (−) ⊗ B applied to our free presentation of A, we see that this cokernel is isomorphic to A ⊗ B
since there is an exact sequence
F1 ⊗ B → F0 ⊗ B → A ⊗ B → 0.
In particular, Tor0,F (A, B) is independent of the choice of the free presentation. We will show below
that the same is true for Tor1,F (A, B) which already partially justifies the following definition. For
now, let us observe that by definition we have
Tor1,F (A, B) = H1 (F ⊗ B) ∼
= ker(F1 ⊗ B → F0 ⊗ B).
Definition 4. Let A and B be abelian groups. The torsion product Tor(A, B) of A and B is
the abelian group
Tor(A, B) = Tor1,F (A, B)
for some choice of a free presentation F → A.
Before we show that the torsion product is well-defined, let us calculate some examples.
Example 5.
(1) Let A be a free abelian group, then 0 → 0 → A → A → 0 is a free presentation
of A. It follows that Tor(A, B) ∼
= 0 for all abelian groups B.
m
(2) Let us consider the abelian group Z/mZ and its free presentation 0 → Z → Z → Z/mZ → 0.
If we use this to calculate Tor(Z/mZ, B) then we have to determine the kernel of
m⊗id
Z ⊗ B → Z ⊗ B.
But under the isomorphism Z ⊗ B ∼
= B this map becomes identified with m : B → B. Thus,
Tor(Z/mZ, B) ∼
= {b ∈ B | mb = 0}
is isomorphic to the group of m-torsion elements in B, justifying the name torsion product.
It can be shown that the torsion product is symmetric and additive in both variables. This allows
us to compute the torsion products of all finitely-generated abelian groups. Let us show now that
the torsion product is well-defined, i.e., that up to canonical isomorphism it is independent of the
chosen free presentation. This will also allow us to conclude that the torsion product is functorial
in both variables. As a first step, let us establish the following fundamental lifting property for free
abelian groups.
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Lemma 6. Let F be a free abelian group and let 0 → B 0 → B → B 00 → 0 be a short exact sequence
of abelian groups. Then we obtain an induced short exact sequence
0 → HomZ (F, B 0 ) → HomZ (F, B) → HomZ (F, B 00 ) → 0.
Proof. One can check that 0 → HomZ (F, B 0 ) → HomZ (F, B) → HomZ (F, B 00 ) is exact even without
imposing any conditions on F (see the exercises). Thus it remains to establish the surjectivity of
the induced map HomZ (F, B) → HomZ (F, B 00 ). But for this purpose let us choose a basis {fj }j∈J
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SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
of F . Given a map g : F → B 00 we want to show that it factors through B, i.e., that there is a
dashed morphism in
B
|>
|
p
|
|
F g / B 00
making the diagram commute. For this it suffices to choose preimages of g(fj ), j ∈ J, under
p : B → B 00 and to use the universal property of free abelian groups, i.e., to extend it linearly. With this preparation we can now establish the following special case of the ‘fundamental lemma
of homological algebra’.
Lemma 7. Let 0 → F1 → F0 → A → 0 be a free presentation and let 0 → G1 → G0 → B → 0 be
a short exact sequence. Any homomorphism h−1 : A → B can be extended to a morphism of short
exact sequences by dashed morphisms h0 and h1 making the following diagram commute:
/0
/ F0
/A
/ F1
0
h−1
h1
h0
/0
/ G1
/ G0
/B
0
Moreover, the resulting map of chain complexes h : F → G is unique up to chain homotopy.
Proof. The proof is left as an exercise.
Corollary 8. Every abelian group has a free presentation and any two free presentations are chain
homotopy equivalent.
Proof. The existence was already established earlier. If we have two free presentations F and F 0
of the same abelian group A, it suffices to apply the previous lemma twice to the identity map
h−1 = idA : A → A. That way we obtain comparison maps h : F → F 0 and h0 : F 0 → F . But their
respective compositions give us comparison maps h0 h : F → F and hh0 : F 0 → F 0 which extend
the identity morphism idA : A → A. In both cases also the identity chain maps idF : F → F and
idF 0 : F 0 → F 0 would be such extensions. By the uniqueness up to chain homotopy for such chain
maps, again by the previous lemma, it follows that h and h0 are inverse chain homotopy equivalences
as intended.
As a consequence we obtain the functoriality of the torsion products.
Corollary 9. The torsion product defines a functor Tor : Ab × Ab → Ab which is additive in both
variables.
Proof. The functoriality in the second variable is immediate since it is simply a composition of
tensor products and homology. The functoriality in the first variable follows from the fundamental
lemma of homological algebra. More specifically, let f : A → A0 be a group homomorphism. Then,
we can choose free presentations F → A and F 0 → A0 . The fundamental lemma guarantees the
existence of a chain map F → F 0 and it suffices to define Tor(f, B) : Tor(A, B) → Tor(A0 , B) as the
induced map
Tor(f, B) = H1 (f ⊗ idB ) : H1 (F ⊗ B) → H1 (F 0 ⊗ B).
This map is independent of the choice of the chain map since all such maps are chain homotopic.
Moreover, the same argument also implies that this assignment is compatible with compositions
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
5
and identities. It is now easy to check that the functorialities in the first and the second variable are
compatible with each. Finally, the functor is additive in each variable because it preserves direct
sums in each variable (see the exercises).
As a final preparation for Theorem 11 let us mention the following particular instance of the
so-called ‘horse-shoe lemma’.
Lemma 10. Let 0 → A0 → A → A00 → 0 be a short exact sequence of abelian groups and let
0 → F10 → F00 → A0 → 0, 0 → F100 → F000 → A00 → 0 be free presentations. Then there is a free
presentation 0 → F1 → F0 → A → 0 and dashed arrows making the following diagram commutes
0
0
0
0
/ F10 _ _ _/ F1 _ _ _/ F100
/0
0
/ F00 _ _ _/ F0 _ _ _/ F000
/0
0
/ A0
/A
/ A00
/0
0
0
0
and such that the rows are exact.
Proof. This proof is left as an exercise.
Theorem 11. Let 0 → A0 → A → A00 → 0 be a short exact sequence of abelian groups and let B
be an abelian group. Then there is natural six-term exact sequence of abelian groups
0 → Tor(A0 , B) → Tor(A, B) → Tor(A00 , B) → A0 ⊗ B → A ⊗ B → A00 ⊗ B → 0.
Proof. Let us begin by choosing free presentations F 0 → A0 and F 00 → A00 . Using the horseshoe lemma we can find a free presentation F → A and a short exact sequence of two-term chain
complexes 0 → F 0 → F → F 00 → 0. Since F 00 is levelwise free, this short exact sequence splits
levelwise. Thus, an application of the additive functor (−) ⊗ B yields a further such short exact
sequence
0 → F 0 ⊗ B → F ⊗ B → F 00 ⊗ B → 0
of two-term chain complexes. It now suffices to consider the associated long exact sequence in
homology. In fact, we saw already that H0 (F ⊗ B) ∼
= A ⊗ B while H1 (F ⊗ B) coincides with
Tor(A, B) by definition, and we thus obtain the desired six-term exact sequence.
Before we return to topology let us make a few more comments about the torsion product. First,
in the definition of the torsion products we decided to replace the first factor by a free presentation
and then continue with our construction. The reader might wonder what happens if instead we
replace the second variable by a free presentation and then continue similarly. It turns out that
in both cases we get naturally isomorphic torsion product functors so that we are free to choose.
Second, in the definition of the tensor product we chose to work with free presentations. The
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SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
motivation for this stems from our earlier observation that the tensor product is well-behaved on
free abelian groups. But this also suggests a different approach. Let us say that an abelian group A
is flat if the functor A ⊗ (−) is exact. Associated to this we have the notion of a flat presentation
of an abelian group and it can be shown that one again obtains a naturally isomorphic torsion
product functor if one works with flat presentations as opposed to free ones. Finally, let us mention
that an abelian group is flat if and only if it is torsion-free, i.e., there are no non-trivial elements of
positive order. This immediately gives us the following example.
Example 12. Let A and B be abelian groups. If A or B is torsion-free then we have Tor(A, B) ∼
= 0.
This applies, in particular, if one of the groups is Q, R, or C.
In fact, in the case that A is torsion-free it suffices to consider the flat presentation
id
0→0→A→A→0
and to conclude by the previous claims.
After this detour through the theory of torsion products, we come back to our original problem
of expressing singular homology with coefficients in terms of integral singular homology and the
coefficients only. We obtain this by first establishing an algebraic fact about chain complexes of
abelian groups which we will then specialize to our geometric situation. Let us agree that for
C ∈ Ch≥0 (Z) the groups B−1 (C), Z−1 (C), H−1 (C) are trivial.
Theorem 13. (Algebraic universal coefficient theorem for homology)
Let C ∈ Ch≥0 (Z) be levelwise free and let A ∈ Ab. Then there is a natural short exact sequence
0 → Hk (C) ⊗ A → Hk (C ⊗ A) → Tor(Hk−1 (C), A) → 0,
k ≥ 0,
in which the first map is given by [zk ] ⊗ a 7→ [zk ⊗ a]. Moreover, there is a splitting of this short
exact sequence (which however is not natural).
Proof. To begin with let us observe that for every k ≥ 1 we have a short exact sequence
j
p
0 → Bk−1 (C) → Zk−1 (C) → Hk−1 (C) → 0
abelian groups. This is just the definition of homology of a chain complex. Now, C is assumed
to be levelwise free so that also the boundaries Bk−1 (C) and cycles Zk−1 (C) are free as they are
subgroups of Ck−1 . Thus, the above short exact sequence is a free presentation of Hk−1 (C) and it
hence qualifies for the construction of Tor(Hk−1 (C), −).
For the rest of the proof, we will use short hand notations like Zk−1 , Z∗−1 and similarly for
boundaries, dropping the dependence on C from notation. As a next step, let us consider the
following additional short exact sequence associated to our chain complex C,
i
∂0
0 → Zk → Ck → Bk−1 → 0,
where ∂ 0 is essentially just the differential ∂ of C with the target changed to its image in order
to obtain a surjection. This short exact sequence splits since Bk−1 is a free abelian group. If we
assemble these groups together for the various k, then we obtain a short exact sequence of chain
complexes
0 → Z∗ → C∗ → B∗−1 → 0
in which we turn the graded abelian groups Z∗ and B∗−1 into chain complexes by endowing them
with the zero differentials. Moreover, this short exact sequence is levelwise split and the additive
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
7
functor (−) ⊗ A hence yields a further levelwise split short exact sequence of chain complexes
∂ 0 ⊗id
i⊗id
0 → Z∗ ⊗ A → C∗ ⊗ A → B∗−1 ⊗ A → 0.
Note that the first and the third chain complexes still carry the trivial differentials. A typical part
of the associated long exact homology sequence looks like
δk+1
δ
k
. . . → Hk+1 (B∗−1 ⊗ A) → Hk (Z∗ ⊗ A) → Hk (C∗ ⊗ A) → Hk (B∗−1 ⊗ A) →
Hk−1 (Z∗ ⊗ A) → . . .
But by the triviality of two of the differentials, this sequence can be rewritten as
δk+1
δ
k
. . . → Bk ⊗ A → Zk ⊗ A → Hk (C∗ ⊗ A) → Bk−1 ⊗ A →
Zk−1 ⊗ A → . . .
and hence gives rise to a short exact sequence
0 → cok(δk+1 ) → Hk (C∗ ⊗ A) → ker(δk ) → 0.
As a next step, let us give a more explicit description of the connecting homomorphism δk in this
long exact sequence. Recall from the general definition of the connecting homomorphism of the
long homology sequence that we have to consider a diagram of the following shape:
∂ 0 ⊗id
Ck ⊗ A
/ Bk−1 ⊗ A
∂⊗id
Zk−1 ⊗ A
i⊗id
/ Ck−1 ⊗ A
Thus, we identify the connecting homomorphism as ‘δk = (i ⊗ id)−1 (∂ ⊗ id)(∂ 0 ⊗ id)−1 ’ which is
simply given by the map
δk = j ⊗ id : Bk−1 ⊗ A → Zk−1 ⊗ A.
Thus, to say it in words, these connecting homomorphisms are the tensor products of the maps
showing up in a free presentation of H∗−1 (C) with the identity on A. It follows from our above
discussion of torsion products that we obtain isomorphisms
cok(δk+1 ) ∼
= Hk (C) ⊗ A
and
ker(δk ) ∼
= Tor(Hk−1 (C), A).
Assembling these things together we finally obtain our intended short exact sequence
0 → Hk (C) ⊗ A → Hk (C ⊗ A) → Tor(Hk−1 (C), A) → 0
in which the first map is given by [zk ] ⊗ a 7→ [zk ⊗ a].
If we apply this theorem to the singular chain complex of a space, then we obtain the following
result.
Corollary 14. (Universal coefficient theorem in singular homology)
Let X be a topological space and let A be an abelian group. Then there is a natural short exact
sequence
0 → Hk (X) ⊗ A → Hk (X; A) → Tor(Hk−1 (X), A) → 0.
Moreover, this short exact sequence splits (but not naturally).
Proof. This is indeed an immediate consequence of the previous theorem applied to the levelwise
free, integral singular chain complex C(X) of the space X.
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SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
There is a similar result for pairs of spaces which is obtained from the algebraic universal coefficient theorem by an application to the relative singular chain complex C(X, Y ) (which is also
levelwise free).
Warning 15. Let us comment on the fact that the splitting of this short exact sequence is not
natural. Given a space X, the existence of a splitting implies that Hk (X; A) is isomorphic to the
direct sum (Hk (X) ⊗ A) ⊕ Tor(Hk−1 (X), A). For each choice of such a splitting we obtain such an
isomorphism and the point is that this cannot be done uniformly for all spaces. Thus, although for
a single space we can express its singular homology with coefficients in the above terms, this cannot
be done in a way compatible with respect to morphisms of spaces. At the end of this section we
will give an example for this phenomenon.
In the case that Tor(Hk−1 (X), A) vanishes, then the universal coefficient theorem implies that
there is a natural isomorphism Hk (X) ⊗ A ∼
= Hk (X; A).
Corollary 16. Let A be a torsion-free abelian group. Then there is a natural isomorphism
Hk (X) ⊗ A ∼
= Hk (X; A), X ∈ Top.
This applies, in particular, to rational, real, and complex singular homology.
As more specific examples we have the following calculations.
Example 17. Let A be an abelian group. Then the homology of the positive-dimensional spheres
with coefficients in A is given by
A , k = 0, n,
Hk (S n ; A) ∼
=
0 , otherwise.
For the complex-projective spaces we obtain
A
Hk (CP n ; A) ∼
=
0
, k = 2l, 0 ≤ l ≤ n,
, otherwise.
The dimensions which show up in the following corollary make sense since, by Lemma 6, the
homology groups under consideration are vector spaces over the respective fields. Recall that a
space X is of finite type if Hk (X) is finitely generated for all k ≥ 0. Since singular and cellular
homology groups are isomorphic, this class of spaces includes CW complexes with only finitely
many cells in each dimension.
Corollary 18. Let X be a space of finite type. Then the k-th Betti number βk (X) = rk(Hk (X))
coincides with the following integers:
βk (X) = dimQ Hk (X; Q) = dimR Hk (X; R) = dimC Hk (X; C).
Remark 19. We include a comment on the universal coefficient theorem. Let f : X → Y be a map
of spaces which induces isomorphisms f∗ : Hk (X) → Hk (Y ) for all k ≥ 0. Then the naturality of
the short exact sequence implies that for any abelian group A we obtain a commutative diagram
0
/ Hk (X) ⊗ A
0
/ Hk (Y ) ⊗ A
/ Hk (X; A)
f∗
/ Hk (Y ; A)
/ Tor(Hk−1 (X), A)
/0
/ Tor(Hk−1 (Y ), A)
/ 0.
Since tensor products and torsion products are functors the two outer vertical maps are isomorphisms and hence the same is true for the map in the middle by the 5-lemma. Thus, if f : X → Y is
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
9
an integral homology isomorphism, then f induces isomorphisms in homology with arbitrary
coefficients. In particular, such a map is also a rational homology isomorphism, a real homology
isomorphism, and a complex homology isomorphism. In general, there is no converse to these
statements, indicating that integral homology groups are more refined algebraic invariants.
In the previous section we saw that singular homology with coefficients satisfies the Eilenberg–
Steenrod axioms. Hence, we expect that there is a variant of cellular homology with coefficients
which we expect to be isomorphic to singular homology with coefficients. This is indeed the case,
but we will only include a short discussion of this. Recall that given a CW complex X, then we
have
M
Ckcell (X) = Hk (X (k) , X (k−1) ) ∼
Z
=
Jk
(k)
where X
denotes the k-skeleton of X and Jk is an index set for the k-cells in X. If A is an
abelian group, then the relative universal coefficient theorem implies that there are isomorphisms
M
M
Hk (X (k) , X (k−1) ; A) ∼
Z⊗A∼
A.
= Hk (X (k) , X (k−1) ) ⊗ A ∼
=
=
Jk
Jk
In fact, the relevant torsion product vanishes since the relative singular homology group under
consideration is free. Thus, setting
Ckcell (X; A) = Hk (X (k) , X (k−1) ; A),
k ≥ 0,
and using again the differential induced from the corresponding long exact sequences of triples we
obtain the cellular chain complex C cell (X; A) with coefficients in A which looks very similar
to what we are used to from integral coefficients. Its homology
Hkcell (X; A) = Hk (C cell (X; A)),
k ≥ 0,
is the cellular homology with coefficients in A. The cellular boundary homomorphism again
admits a geometric description. Using the fact that the isomorphism Z ⊗ A ∼
= A is given by
multiplication, we observe that the boundary homomorphism is basically given by multiplication
with the degrees of certain maps derived from the attaching maps. One establishes the following
theorem.
Theorem 20. (Singular and cellular homology with coefficients are isomorphic.)
Let X be a CW complex and let A be an abelian group. Then there is an isomorphism
Hk (X; A) ∼
= H cell (X; A), k ≥ 0
k
which is natural with respect to cellular maps.
We now want to illustrate that the splitting in the universal coefficient theorem (Corollary 14) is
not natural with respect to maps of spaces. For this purpose, we consider the real projective spaces
RP n . Recall the explicit description of the CW structure on RP n coming with one k-cell in each
dimension 0 ≤ k ≤ n. By the explicit description of the attaching maps, the integral cellular chain
complex C∗cell (RP n ) is given by
2
0
2
0
. . . → Z → Z → Z → Z → Z.
From this it is immediate that C∗cell (RP n ; Z/2Z) looks like
0
0
0
. . . → Z/2Z → Z/2Z → Z/2Z → Z/2Z.
Since all differentials are trivial we deduce the following.
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SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR HOMOLOGY
Example 21. For the real-projective space RP n there are isomorphisms Hk (RP n ; Z/2) ∼
= Z/2Z
for 0 ≤ k ≤ n. In particular, for n = 2 we have H2 (RP 2 ; Z/2Z) ∼
= Z/2Z while H2 (RP 2 ; Z) ∼
= 0.
Thus, a homology group with coefficients can be non-trivial although the corresponding integral
homology group is trivial.
We next consider the quotient map q : RP 2 → RP 2 /RP 1 which collapses the 1-skeleton of RP 2 .
The target of this collapse map is homeomorphic to S 2 , and if we endow S 2 with the CW structure
with a unique 0-cell and 2-cell only, then q∗ : C2cell (RP 2 ) → C2cell (S 2 ) is an isomorphism. It is a
consequence of the naturality statement in Theorem 20 that the induced map
q∗ : H2 (RP 2 ; Z/2Z) → H2 (S 2 ; Z/2Z)
is also an isomorphism, and this can be used to conclude that the splitting in the universal coefficient
theorem is not natural. In fact, if the splitting were natural then the following diagram would
commute
∼
= /
H2 (RP 2 ) ⊗ Z/2Z ⊕ Tor(H1 (RP 2 ), Z/2Z)
Z/2Z ∼
= H2 (RP 2 ; Z/2Z)
q∗ ∼
=
q∗ ⊕q∗ =0
Z/2Z ∼
= H2 (S 2 ; Z/2Z)
∼
=
/ H2 (S 2 ) ⊗ Z/2Z ⊕ Tor(H1 (S 2 ), Z/2Z).
But this is impossible because the vertical map on the left is an isomorphism while the vertical map
on the right has to be trivial by our calculations of these homology groups.
We close this section with a few short comments about possible generalizations. In this section,
we studied a special case of a so-called derived functor. The tensor product is a right exact functor
which is, in general, not left exact, but we managed to define an additional functor which measures
in a systematic way the deviation from this left exactness. This functor is given by the torsion
product Tor(−, −) = TorZ1 (−, −) and it is referred to as the (first left) derived functor of the tensor
product. There is also a tensor product for modules over a general commutative ring R which
has similar formal properties as the one for abelian groups. However, in this more general context
additional complications occur. Note that in the proof of the existence of free presentations of
abelian groups we used the fact that subgroups of free abelian groups are again free abelian. A
corresponding statement for more general commutative rings does not hold true and it is due to
this kind of facts that we have to work with so-called resolutions as opposed to presentations. As
a consequence, there are then higher derived functors TorR
i (−, −) for i ≥ 1 but for our purposes
we do not have to get into this. Instead we refer the reader to any book on Homological Algebra.
Let us only mention that the techniques used in this section work in a bit more general situations.
For example, if we consider modules over principal ideal domains then everything can be redone in
literally the same way.