THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
MARTIN FLUCH
Abstract. Based on the book Foundation of Algebraic Topology by Eilenberg
and Steenrod [ES52] the essentials about the construction of Čech homology
theory will be described.
1. Admissible Categories for Homology Theories
A pair of shall (X, A) shall mean a set X and a subset A ⊂ X. If A = ∅ is the
empty set, then by abuse of notation shall the pair (X, ∅) be abbreviated by X.
Usually X will be equipped with a topology and A will be seen as a subspace of X.
A map of pairs f : (X, A) → (Y, B) shall be a single valued function f from
X to Y such that f (A) ⊂ B. If (X, A) and (Y, B) are both topological pairs,
then f as a map between topological pairs is always assumed to be continuous. If
f : (X, A) → (Y, B) is a map of pairs, then f |A denotes the restriction of f to A,
which is then a map f |A: A → B.
Assume that A is a category where the objects are topological pairs and the
morphisms are maps between such pairs. In order that we can do homology in a
sensible way we need to make some assumptions on the objects and morphisms in
this category. We shall define four conditions which a category of topological pairs
should satisfy in order to considered admissible for homology theories:
(A1 ) Let (X, A) ∈ A. Then the collection of topological pairs
{(∅, ∅), (A, ∅), (X, ∅), (A, A), (X, A), (X, X)}
together with all possible inclusions between those pairs is called the lattice
of the pair (X, A) which can be visualized as follows
(X, ∅)
H
*
(∅, ∅)
HH
j
- (A, ∅)
(X, A)
HH
j
H
- (X, X)
*
2006-01
(A, A)
were all the arrows are inclusions. Note that this lattice also contains all
the identities. If the category A contains all the pairs and inclusions of the
lattice for any pair (X, A) ∈ A, then A is said to satisfy the condition A1 .
(A2 ) If f : (X, A) → (Y, B) is a map of pairs, then in a natural way f induces
maps of pairs from the members of the lattice of the pair (X, A) to the
corresponding members of the lattice of the pair (Y, B). If for any f ∈ A
all those induced maps of pairs are in A, too, then A is said to to satisfy
the condition A2 .
(A3 ) If (X, A) is a topological pair and I = [0, 1] is the closed unit interval, then
(X, A) × I denotes the topological pair (X × I, A × I). We say that A
Date: February 14, 2008.
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MARTIN FLUCH
satisfies the condition A3 if for any (X, A) ∈ A the pair (X, A) × I ∈ A and
the maps
g0 , g1 : (X, A) → (X, A) × I
defined by g0 (x) := (x, 0) and g1 (x) := (x, 1) are in A.
(A4 ) And finally we say that A satisfies the condition A4 if there exists a singelton
space P0 ∈ A, and if for any singelton space P ∈ A, any space X ∈ A and
any map f : P → X holds that f ∈ A, too.
Definition 1 (Admissible Category for Homology). A category A of topological
pairs and maps between them is said to be admissible for homology theory (or just
addmissible if there is no danger of confusion) if it satisfies the conditions A1 , A2 ,
A3 and A4 .
Examples.
(1) The category A1 of all topological pairs and maps between
those pairs is an admissible category. Every other admissible category will
be a subcategory of A1 .
(2) Furthermore the category AC of all compact1 pairs and continuous maps
between them is admissible.
(3) Finally the category ALC of all pairs (X, A) where X is a locally compact
space, A is a closed subset of X and all proper maps between those pairs
(recall that a map is called proper if the pre-image of every compact set is
compact).
2. The Axioms for Homology
Given a category A which is admissible for homology theory we shall consider
three functions:
(1) The first function assigns to each pair (X, A) ∈ A and each k ∈ Z an abelian
group Hk (X, A), the k-th homology group of X modulo A.
(2) The second function assigns to each map f : (X, A) → (Y, B) and each
k ∈ Z a group homomorphism
f∗ : Hk (X, A) → Hk (Y, B)
called the homomorphism induced by f .
(3) The third function assings each pair (X, A) and each integer k ∈ Z a homomorphism
∂: Hk (X, A) → Hk−1 (A)
called the boundary operator.
We shall say that these three functions define a homology theory H = (H, f∗ , ∂)
on A if the following seven axioms (also known as the Eilenberg–Steenrod axioms
for homology) are satisfied:
Axiom 1. If f is the identity then so is f∗ .
Axiom 2. (gf )∗ = g∗ f∗ .
Axiom 3. ∂f∗ = (f |A)∗ ∂.
Axiom 4 (Exactness Axiom). For every pair (X, A) ∈ A the descending sequence
∂
i
j∗
∂
i
∗
∗
. . . −→ Hk (A) −→
Hk (X) −→ Hk (X, A) −→ Hk−1 (A) −→
...
is exact where i∗ and j∗ are the homomrphisms induced by the inclusions i: A → X
and j: X → (X, A).
1Note that compactness shall always imply Hausdorff in this text.
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
3
Axiom 5 (Homotopy Axiom). If f, g: (X, A) → (Y, B) are homotopic maps then
f∗ = g∗ .
Axiom 6 (Excision Axiom). Let (X, A) ∈ A. If U ⊂ X is an open subset of X
such that U ⊂ Int A and such that the inclusion map i: (X \ U, A \ U ) → (X, A) is
admissible, then the induced homomorphism i∗ : Hk (X \ U, A \ U ) → Hk (X, A) is
an isomorphism for all k ∈ Z.
Axiom 7 (Dimension Axiom). If P ∈ A is singelton space, then Hk (P ) = 0 for all
k 6= 0.
3. Simplicial Complexes
This and parts of next section are alike to Spaniers book on Algebraic Topology [Spa66].
Definition 2. Given an arbitrary set V a simplicial complex K with vertices V is
a set of finite non-empty subsets of V which satisfies the following properties:
(1) {v} ∈ K for every v ∈ V , and
(2) if s ∈ K and s0 ⊂ s is a nonempty subset, then s0 ∈ K.
An element s ∈ K is called a simplex and an element v ∈ V is called a vertex. If
s0 ∈ K with s0 ⊂ s, then s0 said to be a face of s. The dimension of a simplex s
is defined to be dim s := |s| − 1. If there exists a least integer n ≥ −1 such that
dim s ≤ n for all s ∈ K, then the dimension of K is n. If no such integer exists
then K is said to be infinite dimensional. In symbols dim K = n or dim K = ∞
respectively.
Let V 0 be another arbitrary set, then any function f : V → V 0 induces a function
f : P(V ) → P(V 0 ) in a canonical way by f (s) := {f (v) : v ∈ s}.
Definition 3. Let K be a simplicial complex with vertices V and K 0 a simplicial
complex with vertices V 0 . Then a function f : K → K 0 is a simplicial map if there
exists a function f 0 : V → V 0 on the set of vertices such that the induced function
f 0 : P(V ) → P(V 0 ) coincides on K with f .
If K is a simplicial complex and L a subset of K, then L is said to be a (simplical)
subcomplex of K if L is again a simplicial complex by its own. A simplicial pair
(K, L) is a pair of simplical complexes where L is a subcomplex of K.
A map of simplical pairs f : (K, L) → (K 0 , L0 ) is a simplicial map f : K → K 0
such that f (L) ⊂ L0 .
4. Chain Complexes of Ordered Simplices
Given a simplicial complex K we shall construct now the chain complex of ordered
simplices:
By an ordered k-simplex of K we shall mean a (k + 1)-tuple (v0 , . . . , vk ) of (not
necessarily distinct) vertices of K such that the set {v0 , . . . , vk } is a simplex of K.
Thus for any non-empty simplicial complex there exists for every k ≥ 0 at least one
ordered k-simplex.
Now the free abelian group Ck (K) generated by all ordered k-simplices is called
the group of k-chains of K. If L is a subcomplex of K, then Ck (L) is a subgroup
of Ck (K) and thus we can define Ck (K, L) := Ck (K)/Ck (L) which is again a free
abelian group. If L = ∅ then Ck (K) and Ck (K, L) coincide. The elements of
Ck (K, L) are called a k-chain.
Example. Assume that
K = {{v0 }, {v1 }, {v2 }, {v0 , v1 }, {v0 , v2 }, {v1 , v2 }}.
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MARTIN FLUCH
Then there exists 3 ordered 0-simplices of K, namely
{(v0 ), (v1 ), (v2 )}
and there exists 9 ordered 1-simplices of K, namely
{(v0 , v0 ), (v0 , v1 ), (v0 , v2 ), (v1 , v0 ), (v1 , v1 ), (v1 , v2 ), (v2 , v0 ), (v2 , v1 ), (v2 , v2 )}
and even more ordered k-simplices of K (k ≥ 2).
If L ⊂ K is the subcomplex L = {{v1 }, {v2 }, {v1 , v2 }} then its ordered 0simplices are
{(v1 ), (v2 )}
and there exists exists 4 ordered 1-simplices of L, namely
{(v1 , v1 ), (v1 , v2 ), (v2 , v1 ), (v2 , v2 )}.
Thus the group C0 (K, L) is the free abelian group generated by the single generator (v0 ) and C1 (K, L) is the free abelian group generated by the set of five
generators {(v0 , v0 ), (v0 , v1 ), (v0 , v2 ), (v1 , v0 ), (v2 , v0 )}.
We define a boundary operator ∂k : Ck (K) → Ck−1 (K) by specifying its values
on the generators of Ck (K). If (v0 , . . . , vk ) is an ordered k-simplex of K, then
∂(v0 , . . . , vk ) is the (k − 1)-chain
∂k (v0 , . . . , vk ) :=
k
X
(−1)i (v0 , . . . , v̂i , . . . , vk )
i=0
where (v0 , . . . , v̂i , . . . , vk ) denotes the ordered (k − 1)-simplex which is obtained
from (v0 , . . . , vk ) by leaving out the i-th vertex.
It is straight forward to verify that this indeed defines a collection of group homomorphisms ∂k : Ck (K) → Ck−1 (K) such that ∂k−1 ∂k = 0. Furthermore, if L is a
subcomplex of K, then ∂k (Ck (L)) ⊂ Ck−1 (L) and thus the homomorphisms induce
homomorphisms ∂k : Ck (K, L) → Ck−1 (K, L). Those induced homomorphisms satisfy ∂k−1 ∂k = 0 by construction, too.
Definition 4 (Ordered Chain Complex). The ordered chain complex of the simplicial pair (K, L) is the (non-negative) chain complex
C(K, L) := {Ck (K, L), ∂k }
If L = ∅ we use the abbreviation C(K) := C(K, ∅).
If G is an R-module (R being a commutative ring with unit), then G is also an
abelian group and we can form the tensor product Ck (K, L; G) := Ck (K, L) ⊗ G
which is again an abelian group. It follows that any element c ∈ Ck (K, L; G) can
be written in a unique way as a finite sum
X0
c=
s ⊗ gs
s
where the sum is taken over all ordered k-simplices s of K which are not a k-simplex
of L and where the gs are elements of G of which all but finit many are equal 0.
(The prime next to the sum sign shall remind of the finiteness of the sum.)
We can give Ck (K, L; G) the structure of a R-module by defining
X0
rc :=
s ⊗ rgs
s
for every r ∈ R and c ∈ Ck (K, L; G). One verifies that ∂k ⊗ id, which maps every
c ∈ Ck (K, L; G) to
X0
(∂ ⊗ id)(c) :=
∂k (s) ⊗ gs ,
s
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
5
makes
{Ck (K, L; G), ∂k ⊗ id}
into a chain complex. By abuse of notation we shall denote the homomorphis ∂k ⊗id
by ∂k for simplicity.
Definition 5. The chain complex
C(K, L; G) := {Ck (K, L; G), ∂k }
is called the ordered chain complex of the simplicial pair (K, L) with coefficients
in G.
Note that C(K, L; Z) is nothing else than C(K, L) in Definition 4. Note further
that any simplicial map of simplicial pairs f : (K, L) → (K 0 , L0 ) defines by
f (v0 , . . . , vk ) := (f (v0 ), . . . , f (vk ))
a chain map f : C(K, L) → C(K 0 , L0 ) of degree 0.
Now in the usual way we define homology groups and induced maps for the above
defined chain complexes and chain maps between them:
Definition 6. Let (K, L) be a simplicial complex and G an R-module. Then the kcycles of C(K, L; G) is the the R-module Zk (K, L; G) := ker ∂k and the k-boundaries
is the R-module Bk (K, L; G) := im ∂k+1 . The k-th homology group of C(K, L; G) is
the quotient R-module
Hk (K, L; G) := Zk (K, L; G)/Bk (K, L; G)
If G = Z, then we shall ommit the Z in the notation, that is we define Zk (K, L) :=
Zk (K, L; G), Bk (K, L) := Bk (K, L; G) and Hk (K, L) := Hk (K, L; G).
If f : (K, L) → (K 0 , L0 ) is a map of simplicial pairs, then the induced chain map
f : C(K, L) → C(K 0 L0 ) yields a chain map f ⊗ id : C(K, L; G) → C(K 0 , L0 ; G).
By abuse of notation we shall name this induced chain map f . Then f induces a
homomorphism f∗ : Hk (K, L; G) → Hk (K 0 , L0 ; G) for all integers k.
We recall two standard results about chain complexes which will be of importance
later:
Proposition 7. For every short exact sequence
β
α
0 −→ C 0 −→ C −→ C 00 −→ 0
of chain complexes and chain maps of degree 0 one can define for every integer
a homomorphism ∂∗ : Hk (C 00 ) → Hk−1 (C 0 ) by ∂∗ := α−1 ∂β −1 (where ∂ is the
boundary homomorphism of the chain complex C) which has the following property:
If 0 → C̄ 0 → C̄ → C̄ 00 → 0 is another exact sequence of chain complexes and
chain maps of degree 0, and f 0 , f and f 00 three chain maps of degree 0 such that
the diagram
0
- C0
0
?
- C̄ 0
f0
commutes, then ∂∗ f∗00 = f∗0 ∂∗ .
- C
f
?
- C̄
- C 00
- 0
f 00
?
- C̄ 00
- 0
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MARTIN FLUCH
Proposition 8. For every short exact sequence
β
α
0 −→ C 0 −→ C −→ C 00 −→ 0
of chain complexes and chain maps of degree 0 the long homology sequence
∂
α
β∗
∂
α
∗
∗
∗
∗
Hk−1 (C 0 ) −→
...
Hk (C 0 ) −→
Hk (C) −→ Hk (C 00 ) −→
. . . −→
is exact.
If f, g: C → C 0 are two chain maps, then by a chain homotopy D: f ' g we
shall mean a chain map D: C → C 0 of degree +1 such that
D∂ + ∂D = f − g
Two chain maps f and g are said to be chain homotopic if there exists a chain
homotopy D: f ' g. A standard proof shows that chain homotopic maps f and g
induce equal maps in homology, that is f∗ = g∗ .
Two maps f, g: (K, L) → (K 0 , L0 ) of simplicial pairs are said to be contiguous if
f (s) ∪ g(s) is a simplex in K 0 and f (s0 ) ∪ g(s0 ) is a simplex in L0 for each simplex
s ∈ K and each simplex s0 ∈ L. For contiguous maps we have the following result
which is for example proven in [ES52, pp. 164f.]:
Lemma 9. If f, g: (K, L) → (K 0 , L0 ) are contiguous maps of simplicial pairs then
f, g: C(K, L; G) → C(K 0 , L0 ; G) are chain homotopic maps. In particular f∗ =
g∗ : Hk (K, L; G) → Hk (K 0 , L0 ; G) for every integer k.
5. Compact Coefficient Groups
If K is a finite simplicial complex and the coefficient group G is compact, then
there exists a natural way to equip Hk (K, L; G) with a toplogy derived from the
topology of G. This is done as follows (see [ES52, pp. 140ff.]):
Assume that F is a free abelian group generated by n elements, say the generators
are c1 , . . . , cn . Then for each c ∈ F ⊗ G there exist unique elements g1 , . . . , gn ∈ G
such that
n
X
ci ⊗ gi
c=
i=1
If one defines with this notation ϕ(c) := (g1 , . . . , gn ) for every c ∈ F ⊗ G, then this
yields an isomorphism
∼
=
ϕ: F ⊗ G −→ G n
Then this isomorphism can be used to transfer the topology of G n to F ⊗ G. By
this way F ⊗ G becomes a compact group. One can verify that this topology is
independent of the choice of the isomorphism ϕ.
Furthermore it follows that if F 0 is another finitely generated free abelian group
and f : F → F 0 is a homomorphism, then the homomorphism
f ⊗ id : F ⊗ G → F 0 ⊗ G
is continuous (note that this is true even in a more general setting: on could replace
id with any continuous homomorphism h: G → G 0 between compact groups and this
would still yield a continuous homomorphism f ⊗ h: F ⊗ G → F 0 ⊗ G 0 ).
Now if (K, L) is a finite simplicial pair, then the group Ck (K, L) is a finitely
generated free abelian group for every k ∈ Z. Thus by the above considerations the
groups Ck (K, L; G) = Ck (K, L) ⊗ G are compact groups. The boundary homomorphisms ∂k for the chain complex Ck (K, L; G) had been defined to be
∂k ⊗ id : Ck (K, L) ⊗ G → Ck−1 (K, L) ⊗ G
and is therefore a continous group homomorphism.
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
7
Then Bk (K, L; G) = im ∂k+1 is compact as the continuous image of a compact group. In particular – since we assumed that compactness always implies
the Hausdorff condition – Bk (K, L; G) is a closed subgroup of Ck (K, L; G). Moreover Zk (K, L; G) = ker ∂k = ∂k−1 ({0}) is a closed subgroup of the compact group
Ck (K, L; G) and thus itself again compact. Thus
Hk (K, L; G) = Zk (K, L; G)/Bk (K, L; G)
is a compact group as a quotient group of a compact group and a closed subgroup
hereof.
Note that any map of simplicial pairs f : (K, L) → (K 0 , L0 ) induces a chain map
f ⊗ id : Ck (K, L; G) → Ck (K 0 , L0 ; G) which is continuous. Therefore the induced
maps f∗ : Hk (K, L; G) → Hk (K 0 , L0 ; G) are continuous for all k ∈ Z, too.
6. Inverse Systems and Their Limits
A pre-ordered set M = (M, ≤) (that is the relation “≤” is on M is reflexive and
transitive) is said to be a directed set if for every α, β ∈ M there exists a γ ∈ M
such that α ≤ γ and β ≤ γ.
Note that α ≤ β and β ≤ α does not necessarily imply α = β.
Let M 0 = (M 0 , ) be another directed set. We say that M 0 is a directed subset
of M if M 0 is a subset of M in the usual sense and if for every α, β ∈ M , α β
implies α ≤ β.
A directed subset M 0 of M is said to be cofinal if for every α ∈ M there exists
a β ∈ M 0 such that α ≤ β.
A map ϕ: (M, ≤) → (M 0 , ) between directed sets is an order preserving function. That is for every α, β ∈ M , α ≤ β implies that ϕ(α) ϕ(β).
Definition 10 (Inverse System). Let M be a directed set. An inverse system
{X, π} of sets directed by M is a function which assigns to each α ∈ M a set
Xα and which assigns to each pair α ≤ β in M a function παβ : Xβ → Xα (called
projections) such that
(1) παα is the identity on Xα for every α ∈ M , and
(2) παβ πβγ = παγ for all α ≤ β ≤ γ in M .
If all the sets Xα are topological spaces (R-modules, topological groups) and
if all the projections are continuous maps (R-module homomorphisms, continuous
group homomorphisms), then {X, π} is called a inverse system of topological spaces
(R-modules, topological groups).
If we have two inverse systems {X, π} and {X 0 , π 0 } then we can consider maps
between them.
Definition 11 (Map of Inverse Systems). Assume that the inverse system {X, π}
is directed by the directed set M and that the inverse system {X 0 , π 0 } is directed
by M 0 . Then a map of inverse systems
Φ: {X, π} → {X 0 , π 0 }
consists of a map of directed sets ϕ: M 0 → M (note the order of the M 0 and M !)
and a function which assigns to each α0 ∈ M 0 a map
ϕα0 : Xϕ(α0 ) → Xα0 0
8
MARTIN FLUCH
such that the diagram
Xϕ(β 0 )
ϕβ 0
- Xβ0 0
ϕ(β 0 )
0
π0 β
α0
πϕ(α0 )
?
Xϕ(α0 )
?
- X0 0
α
ϕα0
commutes for every α ≤ β in M 0 .
Similarly as in the previous definition, if both {X, π} and {X 0 , π 0 } are inverse
systems of topological spaces (R-modules, topological groups) then all the maps
ϕα0 are assumed to be continuous maps (R-module homomorphisms, continuous
group homomorphism).
Given an inverse system {X, π} directed by M we shall abbreviate by X̃ the
Q
cartesian product Xα . If {X, π} is a inverse system of topological spaces then X̃
is a topological space with the product topology.
Definition 12 (Limits of Inverse Systems). Let {X, π} be a inverse system directed
by M . With the inverse limit (or just limit) of this system we mean the subset
X∞ ⊂ X̃ which consists of all those elements (xα ) ∈ X̃ where παβ (xβ ) = xα for all
α ≤ β in M , in symbols
lim{X, π} := X∞ ⊂ X̃
←−
or in short lim Xα := X∞ .
←−
If {X, π} is an inverse system of topological spaces then X∞ will be equipped
with the topology induced by X̃.
Note that in general the set X∞ can be empty. The limit of an inverse system of
topological spaces is by construction again a topological space. It is obvious that
the limit of a inverse system of R-modules (topological groups) is an R-module
(topological group), and that in this case the limit is necessarily non-empty as the
neutral element is always contained in X∞ .
It follows the following result about the inverse limit of non-empty compact
spaces (see [ES52, p. 217]):
Proposition 13. If {X, π} is an inverse system of non-empty compact topological
spaces, then its limit X∞ is a non-empty and compact space.
Now if X∞ is the limit of the inverse system {X, π} directed by M , then we can
define for every α ∈ M maps πα : X∞ → Xα by πα (x) = xα . These maps are called
projections (as are the maps in the definition of an inverse system) and if {X, π}
is a inverse system of topological spaces (R-modules, topological groups) then the
maps πα turn out to be continuous maps (R-module homomorphisms, continuous
homomorphisms).
The inverse limit has the following universal property: Assume that Y is a set
and that we have a collection of maps ψα : Y → Xα such that for all α ≤ β in
M holds παβ ψβ = ψα . Then there exists a unique map η: Y → X∞ making the
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
9
diagram
Xβ
1
3
ψβ
πβ
p p p p p p p p pp p X∞
Y pP
PP
Q πα
PP
Q
P
ψα PPP Q
Q
PP
s
Q
q
P
β
πα
?
Xα
commute for every α ≤ β in M . If {X, π} is an inverse system of topological spaces,
Y is a topological space and all the ψα are continuous maps then η: Y → X∞ is a
continuous map, too. And exactly analogous results hold for the cases that {X, π}
is an inverse system of R-modules or topological groups.
Applying this to the case that Φ: {X, π} → {X 0 , π 0 } is a map of inverse systems
yields the definition of the limit of a inverse system of maps. Given α0 ∈ M 0 let
ψα0 : X∞ → Xα0 0 be the composite ψα0 := ϕα0 πϕ(α0 ) . Then by the universal property
0
of the inverse limit we know that there exists a unique filler ϕ∞ : X∞ → X∞
such
that the diagram
ϕβ 0
Xϕ(β 0 )
πϕ(β 0 )
Q
k
Q
Q
Q
Q
ϕ(β 0 )
πϕ(α0 )
ϕ∞
p p X0
X∞ p p p p p p p p p p∞
πϕ(α0 )
? +
Xϕ(α0 )
- Xβ0 0
3
πβ 0
0
π0 β
α0
Q
Q
πα0 Q
ϕα0
Q
s ?
Q
- X0 0
α
commutes for all α0 ≤ β 0 in M 0 .
0
is
Definition 14 (Limit of a Map of Inverse Systems). The map ϕ∞ : X∞ → X∞
0
0
named the limit of the map Φ: {X, π} → {X , π }, in symbols
lim Φ := ϕ∞
←−
The first hint at what cofinal subsets M 0 of a direct set M are usefull is given by
the fact that if {X, π} is an inverse system and x, y ∈ lim Xα then x = y if and only
←−
if xα = yα for all α ∈ M 0 . This observation leads to the definition of a subsystem:
Definition 15. Let {X, π} be an inverse system directed by the set M and assume
that M 0 is a directed subset of M . Let {X 0 , π 0 } be the restriction of {X, π} to the
directed set M 0 , then {X 0 , π 0 } is called a subsystem of {X, π}. If M 0 is a cofinal
subset of M , then the subsystem is called a cofinal subsystem.
If {X 0 , π 0 } is a subsystem of {X, π} then the inclusion ϕ: M 0 → M together
with the identities ϕα0 = idα0 : Xϕ(α0 ) → Xα0 0 form a map Φ: {X, π} → {X 0 , π 0 }
and this map is called the injection of the system into the subsystem.
Note that the direction of the “injection” which goes from the system into the
subsystem!
Passing to a cofinal subsystem does not change the limit. That is, one can verify
(see [ES52, p. 220]) the following
Proposition 16. If {X 0 , π 0 } is a cofinal subsystem of {X, π} then the limit ϕ∞ of
the injection Φ: {X, π} → {X 0 , π 0 } is a bijection.
10
MARTIN FLUCH
If {X, π} is an inverse system of topological spaces (R-modules, topological
groups) then ϕ∞ is a homeomorphism (isomorphism, continuous isomorphism).
7. Systems of Descending Sequences
By a descending sequence of groups S = {Gk , fk } we mean a collection of groups
Gk indexed by the integers together with homomorphisms fk : Gk → Gk−1 . If
fk−1 fk = 0 (or in other words im fk ⊂ ker fk−1 ) for all k ∈ Z then S is said to be
of order 2. If furthermore im fk = ker fk−1 for all k ∈ Z, then S is said to be an
exact sequence.
If we say that S is a sequence of R-modules (topological groups) then we imply that all the groups and homomorphisms of the sequence are R-modules and
R-module homomorphisms (topological groups and continuous group homomorphism).
If S 0 = {G0k , fk0 } is another sequence of descending groups, then a map (of degree
0) Ψ: S → S 0 is a collection of group homomorphisms ψk : Gk → G0k such that
fk0 ψk = ψk−1 fk for all integers k.
An extension of the definition of inverse systems and their limits to inverse
systems of descending sequences and their limits is done in a natural way:
Definition 17. By an inverse system of descending sequences {S, π} of R-modules
directed by the directed set M we mean a function which assigns to each α ∈ M a
descending sequence Sα = {Gα,k , fα,k } of R-modules and to each α ≤ β in M a
map παβ : Sβ → Sα of degree 0 such that
β
β
πα,k−1
fβ,k = fα,k πα,k
and such that the {Gα,k , πα,k } form for each fixed integer k an inverse system
directed by M .
Then we can form for each fixed k ∈ Z the inverse system {Gk , πk } directed by
M and the fα,k form a map Fk : {Gk , πk } → {Gk−1 , πk−1 }. We denote by Gk,∞
and fk,∞ the respective limits.
Definition 18. By the limit S∞ of the inverse system of decending sequences of
R-modules we shall understand the descending sequence {Gk,∞ , fk,∞ }, in symbols
lim Sα := S∞
←−
In the previous two definitions, replacing the terms “R-module” and “R-module
homomorphism” by topological groups and continuous homomorphisms yields the
definition of inverse systems of descending sequences of topological groups and their
limits.
The limit sequence of an inverse system of descending sequences of compact
groups is again a descending sequence of compact groups.
If all the sequences of an inverse system of descending sequences are of order 2,
then it can be verified that also the limit sequence is of order 2. But exactness is
not necessarily preserved when passing to the limit. Compactness can help here (as
is proven in [ES52, p. 226]):
Proposition 19. Let {S, π} be an inverse system of descending sequences of compact groups. If each sequence of this inverse system is exact, then also the limit
sequence S∞ is exact.
This result will later turn out to be responsible for the Čech homology theory to
satisfy the exactness axiom for in the category AC and compact coefficient groups.
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
11
8. Covering of Spaces
Let X be a topological space. By a (self-indexed) covering λ of X we shall
understand a set of non-empty open subsets of X such that
[
X=
U
U ∈λ
If the covering λ consists only of finite many subsets of X then λ is said to be finite.
Self indexed coverings would be enough to define Čech homology theory. However
one would run into technical difficulties proving that certain ambiguities in the
definition would only look like ambiguities on the first sight. In order to avoid
these technical difficulties we need a bit more subtle version of coverings. This
leads to the definition of indexed coverings.
Definition 20 (Indexed Coverings). Let X a topological space. By an indexed
covering α of X we shall mean a function α: Vα → P(X) (where Vα is an abstract
index set) which assigns to each v ∈ Vα a subset αv of X such that the set {αv :
v ∈ Vα } is a self-indexed covering of X.
If the index set Vα is finite then the covering is said to be finite.
As indexed and self-indexed coverings are very much alike (indeed one can assign
to each indexed covering in a natural way a self-indexed covering and vice versa) we
shall in the following mainly speak of coverings in general and use the terms “indexed
coverings” and “self-indexed coverings” just when we really need to emphasis this
detail.
Definition 21. Let α be as in the definition before and let β be another covering,
indexed by the set Vβ . A refinement projection p: β → α is a function p: Vβ → Vα
such that βv ⊂ αp(v) for every v ∈ Vβ . If there exists a refinement projection
p: β → α then β is said to refine the covering α.
The class of all indexed coverings of X will be denoted by Cov(X). Note that
Cov(X) is a proper class and not a set since there is no restriction on the index
sets.
Yet we shall treat those classes in the following losely as a sets. In the exact
treatment of this issue one would put an upper limit onto the cardinality of the
index sets of Cov(X) which then turns them into sets but which is large enough to
not change the Čech homology groups. [ES52, p. 238]
One verifies that Cov(X) becomes a directed set with the relation “α ≤ β”
whenever “β refines α”. To see this one needs to observe that if α, β ∈ Cov(X) then
a common refinment γ of α and β can be constructed by setting Vγ to be a suitable
subset of Vα × Vβ and defining γ(v,w) := αv ∩ βw .
The following result – which is a consequence of compactness – will be essential
in what follows:
Lemma 22. If X is a compact then any covering α of X can be refined by finite
covering β.
In particular this means that the set of finite coverings Covf (X) is cofinal in
Cov(X) for compact X.
One can now connect an algebraic object with a given covering. The key to this
is the nerve of a covering. Let α be a covering indexed by Vα and let s ⊂ Vα be a
finite set. Then by the carrier Carα (s) of s we shall understand the intersection
\
Carα (s) :=
αv
v∈s
12
MARTIN FLUCH
Definition 23. Let X be a topological space and A ⊂ X. Moreover let α be a
covering of X indexed by Vα . Then the nerve of α with respect to A is the set Aα ⊂
P(V ) which consists of all those finite subsets s ⊂ Vα such that Carα (s) ∩ A 6= ∅.
From the definition it follows immediately that Aα is a simplicial complex. Moreover, if α is a covering of X and A ⊂ X, then Aα is a subcomplex of Xα and therefore (Xα , Aα ) forms a simplicial pair. Thus we can use the tools from Section 3
to Section 5 and assign homology groups to the simplicial pair (Xα , Aα ). But the
resulting homology groups depend very much on the choice of the covering.
Let α be a covering of X and A a subset hereof. If β is another covering of X
refining α, then any refinement projection p: β → α defines a map of simplicial pairs
p: (Xβ , Aβ ) → (Xα , Aα ) in a natural way. Of course this simplicial map depends
on the choice of the refinement projection. But the differences due to the choice of
the refinement projections are not to big as we have the following result which is
proven in [ES52, p. 235]:
Lemma 24. Any two refinement projections p, q: β → α induces contigous maps
of simplicial pairs p, q: (Xβ , Aβ ) → (Xα , Aα )
In particular this means that any refinement projection p: β → α induces an
unique homomorphism p∗ : Hk (Xβ , Aβ ; G) → Hk (Xα , Aα ; G).
9. The Definition of Čech Homology Theory
The previous section revealed the key point in how to connect the topology of
a pair (X, A) of spaces with algebraic opjects. But in order to define a homology
theory one needs a unique way to assign to a topological pair its homology groups.
Using appropriate inverse limits as introduced in Section 6 will help to get rid of
the ambiguity.
We will construct the Čech homology groups for closed pairs (X, A), that is for
topological pairs (X, A) where A ⊂ X is closed in X. We impose this restriction to
our definition because this way we avoid technical difficulties which hide away the
important ideas of the construction process of Čech homology theory. Thus we will
assume in he following (unless otherwise stated) that A is always a closed subset
of X.
9.1. The Definition of the Čech Groups. Let α ∈ Cov(X) and assume that β is
another covering of X refining α. Then by Lemma 24 we know that any refinement
projection pβα induces homomorphisms
pβα∗ : Hk (Xβ , Aβ ; G) → Hk (Xα , Aα ; G)
for every integer k and that these homomorphisms are independent of the choice of
pβα . Moreover, any refinement projection pα
α : α → α induces necessarily the identity
id = pα
α∗ : Hk (Xα , Aα ; G) → Hk (Xα , Aα ; G)
for all k ∈ Z. And if γ is a covering of X refining β and pγβ : γ → β and pγα : γ → α are
refinement projections then for the induced homomorphisms holds pγα∗ = pβγ∗ pγβ∗ .
Thus
{Hk (Xα , Aα ; G), p∗ }
(∗)
forms an invers system of groups directed by Cov(X) for every integer k.
Definition 25 (Čech Homology Groups). The limit of the above defined inverse
system is called the k-th Čech homology group of X modulo A with coefficients in
G, in symbols
Ȟk (X, A; G) := lim{Hk (Xα , Aα ; G), p∗ }
←−
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
13
Note that if G is a R-module then the groups Ȟk (X, A; G) have the structure of
a R-modules, too. But it is not true in a general setting that if G is a compact
group that then also Ȟk (X, A; G) would be compact.
For the later to be true we need to consider a compact pair (X, A). If (X, A) is
a compact pair, then we know by Lemma 22 that set Covf (X) of all finite coverings is cofinal in Cov(X). For every finite covering α we have that the simplicial
complex Xα is finite and thus by the methodes described in Section 5 the groups
Hk (Xα , Aα ; G) become compact groups. If β is an other finite covering refining α
then any refinement projection pβα : β → α will induce continuous homomorphisms
pβα∗ : Hk (Xβ , Aβ ; G) → Hk (Xα , Aα ; G). Thus for every integer k the collection
{Hk (Xα , Aα ; G), p∗ }
(∗∗)
will be a inverse system of compact groups directed by Covf (X). The limit of each
of them will be a compact group by Proposition 13. Now the limit of the injection
of (∗) into the subsystem (∗∗) is by Proposition 13 a isomorphism and we topologize
Ȟk (X, A; G) by demanding that this isomorphism is also a homeomorphism. Thus
we have the
Proposition 26. If (X, A) is a compact pair and G is a compact group then the
Ȟk (X, A; G) are compact groups for all integers k.
9.2. The Construction of the Induced Maps. Next we will construct for a
given map f : (X, A) → (Y, B) of closed topological pairs the induced homomomorphisms f∗ : Ȟk (X, A; G) → Ȟk (Y, B; G) (k ∈ Z). Therefore we need first to
construct for each map of inverse systems
F∗ : {Hk (Xα , Aα ; G), p∗ } → {Hk (Yα , Bα ; G), p∗ }
Let α be an indexed covering of Y with indexset Vα . Since for every v ∈ Vα
the preimage f −1 (αv ) is an open subset of X (as the continuous preimage of the
open set αv ) we can define a covering α0 of X (with the same indexset Vα ) by
αv0 := f −1 (αv ) for every v ∈ Vα0 := Vα .
Then Xα0 is a subcomplex of Yα and Aα0 is a subcomplex of Bα and we shall
denote by fα the inclusion fα : (Xα0 , Aα0 ) ,→ (Yα , Bα ).
If β is an other covering of Y refining α, then any refinement projection pβα : β → α
0
0
induces a refinement projection pβα0 : β 0 → α0 and we have by construction fα pβα0 =
pβα fβ . When passing to homology the homomorphisms induced by the projections
do not depend on the choice of the projections and we get the commutative diagram
Hk (Xβ 0 , Aβ 0 ; G)
fβ∗
- Hk (Yβ , Bβ ; G)
0
pβ
α0 ∗
?
Hk (Xα0 , Aα0 ; G)
pβ
α∗
?
- Hk (Yα , Bα ; G)
fα∗
for all integers k.
Thus the {fα∗ } forms a map F∗ : {Hk (Xα , Aα ; G), p∗ } → {Hk (Yα , Bα ; G), p∗ } and
its limit is denoted by f∗ := lim F∗ .
←−
Definition 27. The homomorphisms
f∗ : Ȟk (X, A; G) → Ȟk (Y, B; G)
as constructed above are called the homomorphism induced by the map f .
It follows that if G is a R-module, then so are the induced homomorphisms f∗ ,
too. If both (X, A) and (Y, B) are compact spaces and G is a compact group, then
the induced homomorphisms f∗ are continuous homomorphisms as well.
14
MARTIN FLUCH
The following facts are easily verified: If f : (X, A) → (X, A) is the identity then
so is the induced homomorphism f∗ : Ȟk (X, A; G) → Ȟk (X, A; G) for every k ∈ Z.
If f : (X, A) → (Y, B) and g: (Y, B) → (Z, C) are two continuous maps of closed
pairs then
g∗ f∗ = (gf )∗ : Ȟk (X, A; G) → Ȟk (Z, C; G)
for every k ∈ Z.
Thus if G is an R-module we have that Ȟk which assigns to each closed pair
(X, A) of an addmissible category of topological spaces the k-th Čech homology
group Ȟk (X, A; G) and which assigns to each continuous map f : (X, A) → (Y, B)
of closed pairs the induced homomorphism
f∗ : Ȟk (X, A; G) → Ȟk (Y, B; G)
is a covariant functor to the category of R-modules.
If G is compact and each pair (X, A) of the admissible category is compact, then
Ȟk is a covariant functor to the category of compact groups.
9.3. The Boundary Homomorphism. Note that if α is covering of X and β
another covering of X refining α, then any refinement projection pβα : β → α induces
a unique homomorphism pβα∗ : Hk (Aβ ; G) → Hk (Aα , G). It follows we optain the
inverse system
{Hk (Aα ; G), p∗ }
directed by Cov(X) (and not by Cov(A) as the inverse system used to define the
Čech homology groups). Its limit will be denoted by Hk (A; G)X where the index
X shall emphasis the use of Cov(X) as the directing set in the inverse system.
We construct a homomorphism Ȟk (A; G) → Hk (A; G)X as follows: If α ∈
Cov(X) is indexed by the set Vα then denote by VαA the set of all indices v ∈ Vα
such that αv ∩ A 6= ∅. Then
α0 : VαA → P(A), v 7→ αv ∩ A
defines a covering of A and thus α0 ∈ Cov(A). We can now define a map
ϕ: Cov(X) → Cov(A)
by ϕ(α) := α0 for every α ∈ Cov(X) and it follows that this map is order preserving. Now the simplicial complexes Aα and Aϕ(α) can be identified in a natural
way by αv 7→ ϕ(α)v . We shall denote the inverse of this identification by ϕα
which is a simplicial map ϕα : Aϕ(α) → Aα . It follows that if β is a covering of X
which is refining α, then the homomorphisms pβα∗ : Hk (Aϕ(β) ; G) → Hk (Aϕ(α) ; G)
and pβα∗ : Hk (Aβ ; G) → Hk (Aα ; G) induced by any refinement projection pβα is independend of the choice of the refinement projection. Thus we get the following
commutative diagram
Hk (Aϕ(β) ; G)
ϕβ∗
- Hk (Aβ ; G)
ϕ(β)
pβ
α∗
pϕ(α)∗
?
Hk (Aϕ(α) ; G)
?
- Hk (Aα ; G)
ϕα∗
for every α ≤ β in Cov(X). Thus ϕ together with the ϕα∗ defines a map of inverse
systems and its limit ϕ∞ is a homomorphism
ϕ∞ : Ȟk (A; G) → Hk (A; G)X
(∗)
Note that since all the ϕα∗ are infact isomorphisms it follows that this homomorphism is actually an isomorphism.
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
15
Now given two coverings α, β ∈ Cov(X) with β refining α, any refinement projection pβα : β → α yields the following commutative diagram of chain complexes
with exact rows:
- C(Aβ ; G)
- C(Xβ ; G)
- C(Xβ , Aβ ; G)
- 0
0
pβ
α
pβ
α
?
- C(Aα ; G)
0
pβ
α
?
- C(Xα ; G)
?
- C(Xα , Aα ; G)
- 0
Now by Proposition 7 there exists a commutative diagram
Hk (Xβ , Aβ ; G)
∂β
- Hk−1 (Aβ ; G)
pβ
α∗
pβ
α∗
?
Hk (Xα , Aα ; G)
?
- Hk−1 (Aα ; G)
∂α
Thus the collection {∂α } defines a map of inverse system and its limit yields a
homomorphism
∂ 0 : Ȟk (X, A; G) → Hk−1 (A; G)X
for every integer k.
Definition 28. For a given closed pair (X, A) and integer k the boundary homomorphism
∂: Ȟk (X, A; G) → Ȟk−1 (A; G)
is the composite of the homomorphism ∂ 0 : Ȟk (X, A; G) → Hk−1 (A; G)X with the
ϕ−1
∞ : Hk−1 (A; G)X → Ȟk−1 (A; G) where ϕ∞ is the isomorphism defined in (∗).
9.4. The Čech Homology Theory. We can now collect the fruits of our work
with the following result. The ommited proof can be found in [ES52, pp. 236–250].
Theorem 29. Let A be a category of closed topological pairs addmissible for homology theory and let G be a R-module. Lel Ȟ, f∗ and ∂ as defined in Definition 25,
27 and 28. Then the Čech Homology Theory Ȟ := (Ȟ, f∗ , ∂) with coefficients in
G is a homology theory on A which satisfies all the Eilenberg–Steenrod axioms for
homology except for the exactness axiom.
We shall call a homology theory H which satisfies all Eilenberg–Stenrood axioms
except for the exactness axiom a partially exact homology theory. And if we want
to emphasis that H satisfies all Eilenberg–Stenrood axioms, then we may do this
by saying that H is an exact homology theory.
Under certain conditions the Čech homology theory Ȟ is an exact homology
theory: one possibility is if A = AC is the category of compact pairs and if the
coefficient groups G is a compact group, as can seen as follows.
Let α be a finite covering the compact space X. It follows that from the inclusions
i: A → X and j: X → (X, A) we get a short exact sequence
j
i
0 −→ C(Aα ; G) −→ C(Xα ; G) −→ C(Xα , Aα ; G) −→ 0
of chain complexes. By Proposition 8 this yields a long exact homology sequence
∂
i
j∗
∂
i
∗
∗
∗
∗
. . . −→
Hk (Aα ; G) −→
Hk (Xα ; G) −→ Hk (Xα , Aα ; G) −→
Hk−1 (Aα ; G) −→
...
From the proof of Proposition 26 we know that the groups Hk (Aα ; G), Hk (Xα ; G)
and Hk (Xα , Aα ; G) are compact for all integers k. This together with the observation that the long homology sequences as above together with the homomorphism pβα∗ induced by refinement projections yield an inverse system of descending
16
MARTIN FLUCH
sequences of compact groups. The limit of this inverse system is
∂0
i
j∗
∂0
i
∗
∗
...
Ȟk (X; G) −→ Ȟk (X, A; G) −→ Hk−1 (A; G)X −→
. . . −→ Hk (A; G)X −→
which is then by Proposition 19 exact. It follows then, that this sequence is isomorphic to the long sequence Čech homology groups
∂
i
j∗
∂
i
∗
∗
...
Ȟk (X; G) −→ Ȟk (X, A; G) −→ Ȟk−1 (A; G) −→
. . . −→ Ȟk (A; G) −→
where the isomorphism is given by the the identities on Ȟk (X; G) and Ȟk (X, A; G)
and the isomorphism ϕ∞ : Ȟk (A; G) → Hk (A; G)X (as defined during the construction of the boundary homomorphism for the Čech homology theory). Thus we have
shown the validy of the exactness axiom and it follows the
Theorem 30. The Čech homology theory Ȟ with compact coefficients G is an exact
homology theory on the category AC of compact pairs and continuous maps between
them.
10. The Continuity of Čech Homology Theory on AC
Let A be a category admissible for homology theory and assume that A satisfies
the following property: when ever {(X, A), π} is a inverse system of pairs and
projections of A, then the limit (X∞ , A∞ ) is again in A. An example for a category
satisfying this condition is the category AC .
If H is a partially exact homology theory on A and {(X, A), π} is an inverse
system of pairs in A directed by M , then the groups Hk (Xα , Aα ) together with the
β
induced homomorphisms πα∗
: Hk (Xβ , Aβ ) → Hk (Xα , Aα ) form an inverse system
of homology groups
β
{Hk (Xα , Aα ), πα∗
}
for every k ∈ Z. Further the projections πα : (X∞ , A∞ ) → (Xα , Aα ) induce homomorphisms πα∗ : Hk (X∞ , Y∞ ) → Hk (Xα , Aα ) which then form a map of inverse
systems
β
L: Hk (X∞ , A∞ ) → {Hk (Xα , Aα ), πα∗
}.
(Note that the domain of L is actually just a trivial inverse system consisting of the
homology group Hk (X∞ , A∞ ) alone.) The limit `∞ of L is then a homomorphism
`∞ : Hk (X∞ , A∞ ) → lim Hk (Xα , Aα ).
←−
Definition 31. A partially exact homology theory H on A is said to be continuous
if the homomorphism l∞ is an isomorphisms for every inverse system {(X, A), π}
in A and every integer k.
That is, for a continuous homology theory we have
Hk lim(Xα , Aα ) ∼
H (X , A ).
= lim
←−
←− k α α
For the Čech homology theory it is shown in [ES52, pp. 261ff.] the following
Theorem 32. Let G be an R-module or a compact group. Then the Čech homology
theory with G as coefficents is a continuous homology theory on the category of
compact pairs AC .
Example. Let p be a prime number. Let Zp be the group of p-adic integers (which
is a compact, totally disconnected abelian group) and assume that X is a compact
Zp -space. For every integer i denote by Zi the open subgroups of Zp such that
Zp /Zi is isomorphic to the cyclic group Zpi of pi elements. Then the Zi form a
descending sequence
Zp = Z0 ⊃ Z1 ⊃ Z2 ⊃ Z3 ⊃ . . .
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
17
of subgroups. Then the action of Zp makes the quotient spaces X/Zi into Zpi spaces. Together with the canonical projections πij : X/Zj → X/Zi (i ≤ j) they
form an inverse system of compact spaces. It follows that the limit of this system
is homeomorphic to the original space X, that is
X∼
X/Zi .
= lim
←−
Thus we have by the continuity of the Čech homology theory on AC that
Ȟk (X; G) ∼
Ȟ (X/Zi ; G)
= lim
←− k
for any R-module or compact coefficient group G. One can now try to study the
properties of the Ȟk (X; G) by studying the groups Ȟk (X/Zi ; G).
Čech homology theory satisfies a much stronger form of excision. For that we
first need some more definitions:
Definition 33. A map of topological pairs f : (X, A) → (Y, B) is said to be a
relative homemorphism if f induces a homeomorphism X \ A ∼
= Y \ B.
For example the inclusion from the excission axiom is always a relative homeomorphism.
Definition 34. Let A be a category admissible for homology theory and let H
be a partially exact homology theory on A. Then H is said to be invariant under
relative homeomorphisms, if every relaive homemorphism f : (X, A) → (Y, B) of A
induces isomorphisms f∗ : Hk (X, A) → Hk (Y, B) for every integer k.
Using the continuity of the Čech homology theory on the category AC on shows
– see [ES52, pp. 266f.] – the
Theorem 35. The Čech homology theory is continuous on the category AC of
compact pairs.
Example. Recall the Čech one-point compactification process. For this assume
that ω is a point which is not contained in any X ∈ ALC . Then for any locally
compact space X we denote by Ẋ the set Ẋ := X ∪ {ω}. As a space the topology of
Ẋ consists of all the open sets of X and the union of {ω} with the countercompact
sets of X (a set U ⊂ X is said to be countercompact if X \ U is compact). It follows
that Ẋ is a compact space.
If (X, A) is a compact pair, then X \ A is locally compact. It follows that the
identity
id : X \ A → X \ A
extends to a map of compact pairs
f : (X, A) → ((X \ A)· , {ω})
by defining f (x) := ω for all x ∈ A and that this map is a relative homeomorphism
(see [ES52, pp. 269f.]). Thus by Theorem 35 it follows that
f∗ : Ȟk (X, A; G) → Ȟk ((X \ A)· , {ω}; G)
is an isomorphism of groups for every integer k.
18
MARTIN FLUCH
11. Extension of Čech Homology Theory to ALC
In the Section 9 we defined the Čech homology theory for any admissible category
of closed pairs. This definition yields only a homology theory for ALC which fails
to be exact. In order to get a exact homology theory we can start with the Čech
homolohy theory on AC and extend it to ALC in such a way that it coincides with
the Čech homology theory on AC . This extension process will be described in this
section.
If (X, A) is a pair of ALC , then (Ẋ, Ȧ) is a pair in AC . If f : (X, A) → (Y, B)
is an admissible map of pairs in ALC , then this map extends to a continuous map
f˙: (Ẋ, Ȧ) → (Ẏ , Ḃ) by defining f˙(ω) := ω.
˙ on ALC with coefficients G as
We shall define a homology theory Ḣ = (Ḣ, f˙∗ , ∂)
follows:
(1) For any (X, A) in ALC the k-th homology group is
Ḣk (X, A; G) := Ȟk (Ẋ, Ȧ; G).
(2) For any f : (X, A) → (Y, B) in ALC the induced homomorphisms are the
one induced by f˙: (Ẋ, Ȧ) → (Ẏ , Ḃ) in the Čech homology theory Ȟ on AC .
(3) Given a pair (X, A) ∈ ALC the boundary homomorphism
˙ Ḣk (X, A; G) → Ḣk−1 (A; G)
∂:
is defined to be composite of the
i
∂
∗
Ȟk (Ẋ, Ȧ; G) −→ Ȟk−1 (Ȧ; G) −→
Ȟk−1 (Ȧ, {ω}; G)
where ∂ is the boundary homomorphism of the Čech homology theory for
AC and i∗ is the homomorphism induced by the inclusion Ȧ ,→ (Ȧ, {ω}).
Then one can prove (and this is done in [ES52, pp. 269ff.]) the following
˙ defines a partially
Theorem 36. Let G be an R-module. Then Ḣ = (Ḣ, f˙∗ , ∂)
exact homology theory on ALC which is continuous and invariant under relative
homeomorphisms. If G is a compact group then Ḣ is even an exact homology theory
on ALC .
If (X, A) is a compact pair in ALC , then the inclusion i: (X, A) → (Ẋ, Ȧ) is
admissible in AC and a relative homeomorphism. Thus the induced map
i∗ : Ȟk (X, A; G) → Ȟk (Ẋ, Ȧ; G)
is an isomorphism for all integers k and we get
Ḣk (X, A; G) = Ȟk (Ẋ, Ȧ; G) ∼
= Ȟk (X, A; G)
Furthermore, if f : (X, A) → (Y, B) is a map of pairs in AC , then f˙: (Ẋ, Ȧ) → (Ẏ , Ḃ)
is admissible in ALC and one has for every integer k the commutative diagram
Ȟk (X, A; G)
?
Ḣk (X, A; G)
f∗
- Ȟk (Y, B; G)
?
- Ḣk (Y, B; G)
f˙∗
where the vertical arrows are the isomorphisms induced by the inclusions (X, A) ,→
(Ẋ, Ȧ) and (Y, B) ,→ (Ẏ , Ḃ).
THE CONSTRUCTION OF THE ČECH HOMOLOGY THEORY
19
Similarly we have for every compact pair (X, A) and for every integer k the
commutative diagram
Ȟk (X, A; G)
∂-
Ȟk−1 (A; G)
?
Ḣk (X, A; G)
∂˙ -
?
Ḣk−1 (A; G)
where the vertical arrows are the isomorphisms induced by the inclusions (X, A) ,→
(Ẋ, Ȧ) and A ,→ Ȧ.
This justifies that we can identify the groups Ḣk (X, A; G) and Ȟk (X, A; G) by the
isomorphisms induced by the inclusion (X, A) ,→ (Ẋ, Ȧ) and we can consider the
homology theory Ḣ as an extension of Ȟ from the category AC to ALC . Therefore
we will denote the groups Ḣk (X, A; G) by abuse of notation again by Ȟk (X, A; G)
for all pairs (X, A) ∈ ALC .
Example. Recall the induced homomorphism
f∗ : Ȟk (X, A; G) → Ȟk ((X \ A)· , {ω}; G)
from the example in the previous section. By definition we have
Ȟ ((X \ A)· , {ω}; G) = Ḣ (X \ A; G)
k
k
and thus with the identification introduced before we have the isomorphisms
Ȟk (X, A; G) ∼
= Ȟk (X \ A; G)
which are all induced by the continuous map f : (X, A) → ((X \ A)· , {ω}).
Appendix A. The Problem of Cov(X) Being a Proper Class
During the construction of the Čech homology theory we used Cov(X) in a sloopy
way as if it was a set. But since there is no restriction on the index sets used for the
coverings of Cov(X) we have that Cov(X) is proper class and not a set. In [ES52,
p. 238] Eilenberg and Steenrod describe how this problem is avoided:
If M is a set then we denote by CovM (X) the collection of all coverings of
coverings α of X such that the index set Vα is a subset of M . Then CovM (X)
is a set and not a proper class. If N is another set such that M ⊂ N , then
CovM (X) ⊂ CovN (X).
Now in [ES52, p. 238] the following result is proven:
Lemma 37. Let ω(X) be the least cardinal such that the space X has a base for
the open sets of cardinal power ω(X). Then for any pairs of sets M ⊂ N with
card(M ) ≥ ω(X) it follows that CovM (X) is a cofinal subset of CovN (X).
If M and N are two arbitray sets with cardinality at least ω(X). Denote
by Ȟk (X, A; G)M the limit of the inverse system {Hk (X, A; G), p∗ } directed by
CovM (X), and alike the groups Ȟk (X, A; G)N and Ȟk (X, A; G)M ∪N are defined.
Then since both CovM (X) and CovN (X) are cofinal in CovM ∪N (X) it follows that
the isomorphism defined by the injection of the system directed by CovM ∪N (X)
into the subsystems directed by CovM (X) and CovN (X) respectively define isomorphism Ȟk (X, A; G)M ∼
= Ȟk (X, A; G)M ∪N ∼
= Ȟk (X, A; G)N . The composite of
∼
these isomorphisms shall be denoted by iN
:
Ȟ
k (X, A; G)M = Ȟk (X, A; G)N . One
M
M
verifies that iM is the identity and that for any three sets M , N , R with cardinality
N M
at least ω(X) we have iM
R = iR iN . Thus these isomorphisms define a transitive
system of groups (as defined in [ES52, p. 17]) and in this sense Ȟk (X, A; G)M may
20
MARTIN FLUCH
be regarded to be independent of the choice of the set M with card(M ) ≥ ω(X)
and is denoted by Ȟk (X, A; G).
If A is an admissible category such that the cardinals ω(X) for X ∈ A have
an upper bound, then the above discussion for a single pair (X, A) can be applied
to A.
Appendix B. Compactification
In the definition of the one-point compactification we demanded that ω is a point
which is not contained in any space X ∈ ALC . This amounts actually to replacing
ALC by a subcategory and is a highly artificial requirement. The reason for this
requirement is that we wanted that for every pair (X, A) its compactification (Ẋ, Ȧ)
is again a pair.
A possible proper solution – as described in [ES52, p. 293] – is to introduce a
generalizied version of a pair (X, A). A generalized pair is a triple (X, A, ϕ) where
X and A are spaces and ϕ: A → X is an embedding. If X, A and ϕ are in ALC
then we say that (X, A, ϕ) is in ALC . Then the compactification (Ẋ, Ȧ, ϕ̇) is a
generalized couple even if different points are used for the compactification of X
and A.
On might use the Neumann–Bernays–Gödel axiomatics to make a suitable natural choice of the compactification point. In this setting a set maybe an element
of another set but never an element of itself. Thus we may define Ẋ to be the set
Ẋ := X ∪ {X} with the topology defined in the example in the end of Section 10.
References
[ES52] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Princeton
University Press, 1952.
[Spa66] Edwin Henry Spanier. Algebraic Topology. Springer–Verlag, 2nd edition, 1966.