Download LECTURE NOTES 4: CECH COHOMOLOGY 1

LECTURE NOTES 4: CECH COHOMOLOGY
MATTHEW ANDO
1. Presheaves
A topological space X determines a category Open(X) with
obC {U ⊂ X|U is open}
and, if U, V ∈ Open(X) then Open(X)(U, V ) is empty unless U ⊆ V , in which case it consists of a single
element iVU : U → V.
Notice that Open(X) is a subcategory of the category T of topological spaces.
Definition 1.1. Let X be a topological space, and let C be a category. A presheaf on X with values in C
is a contravariant functor
F : Open(X)op → C.
More generally, if E is a category, then a presheaf on E with values in C is just a contravariant functor
F : E op → C.
Example 1.2. Suppose that X and T are topological spaces. If U is an open set, let
T (U ) = map(U, T )
be the set of continuous functions from U to T . For f ∈ T (U ) and V ⊂ U , let
f |V ∈ T (V ).
be the restriction of f to V . This defines a contravariant functor
Open(X) → (sets).
This is an example of a presheaf of sets. It is also an example of a represented functor.
Definition 1.3. Let Z be an object of C. Then the rule
def
X 7→ X Z = C(Z, X)
is a covariant functor from C to (sets). This is the functor represented by Z. The rule
def
X 7→ Z(X) = C(X, Z)
is a contravariant functor from C to (sets). It is the functor corepresented by Z.
Example 1.4. Let A be an abelian group, considered as a space with the discrete topology. Then
A(U ) = map(U, A)
is a group: if f, g : U → A, then their sum is
∆
f ×g
U −→ U × U −−−→ A × A → A.
With this group structure, A is a presheaf of abelian groups on X.
The examples we have studied so far have the feature that they are the restriction to Open(X) of a
presheaf on the category of topological spaces.
Date: February 2007 and February 2008.
c
2007,
2008 by the author. You may copy this document for personal study.
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2. Products and sums
Let {Xα }α∈I be a collection of objects of C. The product of the Xα is an object
a collection of maps
Y
πα
Xα −−→
Xα
Q
α∈I
Xα together with
α∈I
such that, for every collection of arrows
fα
T −→ Xα
in C, there is a unique map f : T →
Q
Xα such that
πα f = fα
for all α ∈ I. This means that
C(T,
Y
Xα ) =
Y
α
C(T, Xα ),
α
where on the right the product is the classsical one, in the category of sets.
`
The sum or coproduct of the Xα is an object α∈I Xα together with a collection of arrows
a
Xα → iα
Xα
α∈I
such that, for every collection of arrows
fα
Xα −→ T
in C, there is a unique map f :
`
Xα → T such that
f iα = fα
for all α ∈ I. This means that
a
Y
map(Xα , T ),
map( Xα , T ) =
α
where again on the right, the product is the classical product of sets.
Exercise 1. Show that the usual notion of product and sum of abelian groups or sets coincides with the
categorical notion. Show that disjoint union is the sum in the category of topological spaces. What is the
sum in the category of pointed topological spaces?
In good cases, contravariant functors convert sums to products.
Example 2.1. Let F be a representable presheaf on C, that is,
F (U ) = C(U, T )
for some T ∈ C. Then
a
a
F ( Xα ) = C( Xα , T )
α
=
Y
C(Xα , T )
α
=
Y
α
2
F (Xα ).
3. Sheaves
Let X be a topological space. For U ∈ Open(X), a cover of U is a collection U = {Uα }α∈I of open subsets
of U such that
[
Uα = U.
α∈I
It is important to distinguish this union from the coproduct
of the Uα in the category of topological spaces,
`
that is, their disjoint union, which will be denoted Uα .
If U is a cover of U , then there is a sequence of spaces
U o
`
o
α0 ∈I
d0
`
Uα0 o
α0 ,α1 ∈I
Uα0 ∩ Uα1
(3.1)
d1
In the diagram, d0 is the map which on the Uα0 α1 summand is the inclusion
a
Uα0 α1 → Uα1 →
Uα ,
α∈I
and d1 is
Uα0 α1 → Uα0 →
a
Uα .
α∈I
That is, di omits the i index.
Let F be a presheaf on X taking values in C. Suppose that F takes coproducts of spaces to products in
C. Applying F to the sequence above gives in C
/
F (U )
Q
/Q
d0
/
α0 ∈I F (Uα0 )
d1
α0 ,α1 ∈I
F (Uα0 ∩ Uα1 ).
(3.2)
As an exercise for yourself, you can check that as long as C has products, you can obtain the sequence
(3.2) from the sequence (3.1) even if F does not convert sums to products.
Definition 3.3. The presheaf F is a sheaf if for all U and all covers U, the sequence (3.2) is an equalizer.
That is, to give a map
T → F (U )
in C is equvalent to giving a map
g:T →
Y
F (Uα0 ),
α0
such that
d0 g = d1 g : T →
Y
F (Uα0 ∩ Uα1 ).
α0 ,α1 ∈I
We mention two convenient variants of this definition.
Ordered cover. A cover Uαα∈I will be called ordered if there is partial ordering on I such that if Uα ∩ Uβ
is non-empty, then either α < β or β < α or α = β. Possibly it is correct terminology to say that if α and β
are incomparable, then Uα ∩ Uβ is empty.
In that case, we can considerable replace the sequence (3.1) and (3.2) above with with
U o
F (U )
/
`
Q
α0
o
Uα0 o
d0
`
α0 <α1
Uα0 ∩ Uα1
(3.4)
d1
d0
α0 F (Uα0 )
d1
3
/
/Q
α0 <α1
F (Uα0 ∩ Uα1 ).
(3.5)
It is easy to check that F is a sheaf if and only if the sequence (3.5) is an equalizer for every ordered cover
{Uα }α∈I of U.
Abelian sheaves. If A is an abelian category, then in the sequence (3.2) or (3.5), we can set
δ = d0 − d1 .
It is easy to check that F is a sheaf if and only if for all covers {Uα }α∈I , the sequence
δ
Y
0 → F (U ) →
Y
F (Uα ) −
→
α0 ∈I
F (Uα0 ∩ Uα1 )
α0 ,α1 ∈I
is exact. You can also consider only ordered covers, and require that the sequence
0 → F (U ) →
Y
δ
Y
F (Uα ) −
→
F (Uα0 ∩ Uα1 )
α0 <α1
α0 ∈I
be exact.
4. Examples
The presheaf of sets
T (U ) = T (U, T )
represented by a space T is a sheaf. If G is a topological abelian group, then
G(U ) = T (U, G)
is a sheaf of abelian groups, since given a, b : U → G we can form their sum
∆
a×b
U −→ U × U −−→ G × G −
→ G.
If G is any old abelian group, then we can consider it as a discrete space. In that case, the sheaf U 7→
G(U ) = T (U, G) is called the constant sheaf with values in G.
The very constant presheaf with values in G is the presheaf G! whose value on an open set U is
G!(U ) = G.
(Check that this is not a sheaf)
Let M be a smooth manifold. The sheaf of smooth functions on M is the sheaf C ∞ whose value on an
open set U is
C ∞ (U ) = smooth functions U → R.
Let V be a smooth complex variety. The sheaf of holomorphic functions on V is the sheaf OV whose value
on an open set U is
OV (U ) = holomorphic functions on U.
In all the examples, the object F (U ) is some sort of set.
Definition 4.1. Suppose that F is a presheaf on X, taking values in a category of sets. Then we say that
F (U ) is the “set of sections” of F over U , and if f ∈ F (U ), then f is a section of F over U .
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5. The Cech complex
Let X be a space, and let U = {Uα }α∈I be a cover of U . For convenience, if α0 , . . . , αp ∈ I, let
p
\
Uα0 ,...,αp =
Uαi
i=0
Notice that the sequence of open sets used to define a sheaf has a natural extension
U o
`
α0
o
d1
Uα0 o
`
d0
α0 ,α1
Uα0 α1 o
oo
d2
d1
d0
`
o
α0 ,α1 ,α2
Uα0 α1 α2 o
o
d3
...
d2
d1
o
d0
The map di is given on the Uα0 ···αp summand by
Y
Uα0 ···αp −
→ Uα0 ···α̂i ···αp −
→
Uβ0 ···βp−1
β0 ···βp−1
We shall often assume that our cover is ordered, and consider instead the smaller complex
U o
`
o
α0 Uα0 o
d1
d0
`
oo
α0 <α1 Uα0 α1 o
d2
d1
d0
`
o
o
α0 <α1 <α2 Uα0 α1 α2 o
o
d3
...
d2
d1
d0
Lemma 5.1. In both the ordered and unordered case, we have
di dj = dj−1 di
whenever i < j.
Proof. The left side is the inclusion
Uα0 ···αp+1 → Uα0 ···α̂j ···αp+1 → Uα0 ···α̂i ···α̂j ···αp+1 .
The right side is the inclusion
Uα0 ···αp+1 → Uα0 ···α̂i ···αp+1 → Uα0 ···α̂i ···α̂j ···αp+1 .
Now suppose that F is a presheaf on X, taking values in a category C with products. Then we have the
“cosimplicial object” C
`
α0
F (Uα0 )
d0
d1
d0
d1
d2
/`
/ α0 <α1 F (Uα0 α1 )
// `
/ α0 <α1 <α2 F (Uα0 α1 α2 )
d0
d12
d
d3
/
/
/ ....
/
This is the (ordered) “Cech cosimplicial object” associated to U, with coefficients in F , and we write
a
def
C p (U; F ) =
F (Uα0 ···αp ).
α0 <···<αp
Lemma 5.1 implies that
dj di = di dj−1
whenever i < j.
If C is an abelian category, then we may define
δ=
p+1
X
(−1)i di : C p (U; F ) → C p+1 (U; F ).
i=0
Lemma 5.3.
δ 2 = 0.
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(5.2)
Proof. Let’s start from C p . Then
δ2 =
p+2
X
(−1)j dj
j=0
=
p+1
X
(−1)i di
i=0
p+2 X
p+1
X
(−1)i+j dj di
j=0 i=0
=
p+2 X
j−1
X
(−1)i+j dj di +
j=0 i=0
p+2 X
p+1
X
(−1)i+j dj di .
(5.4)
j=0 i=j
The left sum in the last line has i < j, so we rewrite it
p+2 X
j−1
X
p+2 X
j−1
X
(−1)i+j dj di =
j=0 i=0
(−1)i+j di dj−1 .
j=1 i=0
Now set u = i and v = j − 1 and this becomes
p+1 X
X
(−1)u+v+1 du dv .
u=0 v≥u
Setting u = j and v = i in the right sum in (5.4) gives
p+2 X
p+1
X
p+1 X
X
(−1)i+j dj di =
(−1)u+v du dv .
u=0 v≥u
j=0 i=j
Thus the two sums in (5.4) cancel, giving the result.
Definition 5.5. Let F be a sheaf on X taking values in an abelian category. The Cech complex of the cover
U with coefficients in F is the cochain complex
δ
δ
δ
C 0 (U; F ) −
→ C 1 (U; F ) −
→ C 2 (U; F ) −
→ ···
The Cech cohomology of U with coefficients in F is the cohomology of this cochain complex,
Ȟ p (U; F ) = H p (C ∗ (U; F ); δ).
Before proceeding, we note that if F takes values in abelian groups, and f ∈ F (Uα0 ···αp ) is a section, then
(δf )α0 ···αp+1 =
p+1
X
(−1)i fα0 ···α̂i ···αp+1 .
i=0
We also note the following.
Proposition 5.6. Let F be a sheaf. Then for any cover U of U ,
∼ F (U ).
Ȟ 0 (U; F ) =
6. Comparison of Cech and singular cohomology
Let X be space, and let U = {Uα }α∈I be an ordered open cover. Let G be an abelian group, considered
as a discrete topological space.
Suppose that (S ∗ , d) is a presheaf of cochain complexes on X, with the following properties.
(1) For each q ≥ 0,
Ȟ 0 (U, S q ) ∼
= SqX
Ȟ p (U, S q ) = 0
p>0
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(2) If U ⊂ X is homotopy equivalent to a discrete set of points, then
H 0 (S ∗ U ; d) ∼
= Gπ0 U = G(U )
H q (S ∗ U ; d) = 0
q > 0.
We shall prove the following.
Proposition 6.1. If the cover U is countable, then the presheaf
S ∗ U = CU∗ (U, G)
of singular cochains which respect U satisfies these hypotheses.
Recall that the method of barycentric subdivision proves that for any cover, CU∗ (U, G) and C ∗ (U, G) are
chain homotopy equivalent, so
H ∗ (CU∗ (U, G)) ∼
= H ∗ (U, G).
Bott and Tu prove that if G = R, X is a smooth manifold, and U is a countable ordered cover, then the
presheaf
S ∗ U = Ω∗ U
of differential forms satisfies these hypotheses.
Suppose that U is a good cover. By this we mean a countable ordered cover with the property that every
finite intersection Uα0 ...αp is empty or contractible. Then we have the following.
Theorem 6.2. With these hypothesis, there is a canonical isomorphism
∼ Ȟ n (U, G).
H n (S ∗ X; d) =
Corollary 6.3. The Cech cohomology of a good cover of X with coefficients in the discrete abelian group G
coincides with the singular cohomology of X with coefficients in G.
7. The double complex
To compare the Cech and singular cohomology, introduce the double complex
C p,q = Č p (U, S q )
Y
=
S q (Uα0 ...αp ).
α0 <···<αp
That is, we have the following.
δ
δ
δ
δ
δ
δ
Č 0 (U; S 2 ) −−−−→ Č 1 (U; S 2 ) −−−−→ Č 2 (U; S 2 ) −−−−→
x
x
x



d
d
d
Č 0 (U; S 1 ) −−−−→ Č 1 (U; S 1 ) −−−−→ Č 2 (U; S 1 ) −−−−→
x
x
x



d
d
d
Č 0 (U; S 0 ) −−−−→ Č 1 (U; S 0 ) −−−−→ Č 2 (U; S 0 ) −−−−→
With our hypotheses, we have
Hδ0,q = S q X
Hδp,q = 0
p > 0,
and so
Hd0,q Hδ = H q (S ∗ X; d)
Hdp,q Hδ = 0
p > 0.
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Let
Tn =
M
C p,q .
p+q=n
with differential
D = δ + (−1)p d.
Our study of double complexes shows that
0,n
H n (T ∗ ) ∼
= Hd Hδ ∼
= H n (S ∗ X; d).
Taking cohomology in vertical direction first, and recalling that U is a good cover, we have
Y
Hdp,0 ∼
Gπ0 Uα0 ···αp ∼
=
= C p (U; G),
α0 <···<αp
and
Hdp,q = 0
for q > 0. Thus
Hδp,0 Hd = Ȟ p (U; G),
while
Hδp,q Hd = 0
for q > 0. Our work on double complexes now easily implies the desired result.
8. Proof of Proposition 8.1
The comparison of singular and Cech cohomology depended on the following result. Let U be a cover of
a space X. Let A be any abelian group, and let S ∗ be the presheaf on X given by the formula
S ∗ U = CU∗ (U ; A).
Proposition 8.1. For all q, the sequence
0 → CUq (X; A) → Č 0 (U; S q ) → Č 1 (U; S q ) → . . .
is exact. Equivalently,
Ȟ 0 (U, S q ) = S q (X)
and
Ȟ p (U, S q ) = 0
for p > 0.
Bott and Tu prove this for countable, ordered, indexed covers U. Here we shall explain why the result is
true for any indexed cover: countability is not required. We do not need to choose an ordering, but we do
need the Axiom of Choice.
It’s simplest to give the proof without assuming countability or a choice of ordering, so we’ll used the Cech
comoplex of an unordered cover. We shall not prove that the Cech complex we discuss here gives the same
cohomology as the one used elsewhere in the course. You can either adapt the formulae here, or convince
yourself that the two complexes have the same cohomology.
So in this section, let U = {Uα }α∈I be an indexed open cover of a space X. For any subspace W of X, let
S(W ) = CqU (W )
denote the singular q-chains on W which respect the cover U. Let
M
Čp =
S(Uα0 ...αp ).
α0 ,...,αp
The sum is over all arrays α0 , . . . , αp of p + 1 elements of I. That is, we do not assume that they are in
increasing order, and they are allowed to repeat.
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Here’s some useful notation, which we use to help us keep track of where we are in the Cech complex.
Suppose that
σ : ∆q → X
is a simplex, which happens to land in Uα0 ...αp . Then we shall write
σ|α0 ...αp
for the element of
M
Cp =
S(Uβ0 ...βp )
β0 ...βp
which is σ in the α0 . . . αp summand, and zero otherwise.
For p > 0, let’s define a boundary operator
δ : Čp → Čp−1
as follows. It suffices to define δ on each summand S(Uα0 ...αp ), and on this summand it suffices to give the
value of δ on a q-simplex
σ : ∆q → Uα0 ...αp .
For such a simplex, we set
p
X
δσ =
(−1)j σ|α0 ...α̂j ...αp .
j=0
2
You can check that δ = 0. We claim the following, without proof.
Proposition 8.2. The homology of this complex is isomorphic to the homology of its ordered indexed analogue. The cohomology of the contravariant analogue of this complex calculates the same Cech cohomology
as its ordered indexed analogue.
Note that there’s also an evident map
δ : Č0 → S(X),
which sends σ : ∆q → Uα to
σ : ∆q → Uα → X.
For convenience, let’s set
Č−1 = S(X).
Proposition 8.3. The complex
δ
δ
δ
δ
− ...
− Č2 ←
− Č1 ←
0 ← S(X) ←
− Č0 ←
is exact. More precisely, for p ≥ −1 there is a homomorphism
K : Čp → Čp+1
such that
δK + Kδ = 1.
You should convince yourself that these results together give the desired result about Cech cohomology.
Proof. The proof proceeds by giving a simple formula for K: in this sense it’s easier than the proof of Bott
and Tu. First, by definition
S(X) = Z{σ : ∆q → X|σ(∆q ) ⊆ Uα for some α ∈ I},
and the map
Č0 → S(X)
is surjective. We choose a section as follows. For each simplex σ as in the definition of S(X), choose an
element
c(σ) ∈ I
9
such that
σ(∆q ) ⊆ Uc(σ) .
Define K : S(X) → Č0 by the formula
Kσ = σ|c(σ)
and extend by linearity. Clearly from S(X) to S(X) we have
δKσ = σ
and so δK = 1.
For p ≥ 0, define K on Čp by the formula
K(σ|α0 ...αp ) = σ|c(σ)α0 ...αp .
Then
δK(σ|α0 ...αp ) = δσ|c(σ)α0 ...αp
= σ|α0 ...αp +
p
X
(−1)j+1 σ|c(σ)α0 ...α̂j ...αp .
j=0
Meanwhile
Kδ(σ|α0 ...αp ) = K
=
p
X
(−1)j σ|α0 ...α̂j ...αp
j=0
p
X
(−1)j σ|c(σ)α0 ...α̂j ...αp .
j=0
One sees easily that Kδ + δK = 1, as required.
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