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Homology Theory
Kay Werndli
13. Dezember 2009
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CONTENTS
Chapter 1
1.
2.
3.
4.
The Eilenberg-Steenrod Axioms
First Consequences . . . . .
Reduced Homology . . . . .
Homology of Spheres . . . .
3
6
9
10
Acyclic Models . . . . . . . . . . . . . . . . . . . . . .
13
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Acyclic Model Theorem . . . . . . . . . . . . . . . . . . . . .
13
14
Chapter 3
1.
2.
3.
4.
5.
Singular Homology
Definitions . . . . . . . .
Homotopy Invariance . . .
Barycentric Subdivision . .
Small Simplices and Standard
Excision . . . . . . . . .
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3
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Chapter 2
1.
2.
Axiomatic Homology Theory . . . . . . . . . . . . . . . .
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Models
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17
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Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Chapter 1
AXIOMATIC HOMOLOGY THEORY
[Der Satz] verhüllt die geometrische Wahrheit mit dem Schleier
der Algebra.
TAMMO TOM DIECK
Homology theory has been around for about 115 years. It’s founding father was the
french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian
groups were the right context to study homology and not the then known and extensively
used Betti numbers. In the decades after the advent of Poincaré’s homology invariants, many
different theories were developed (e.g. simplicial homology, singular homology, Čech homology etc.) by many well-known mathematicians (e.g. Alexander, Čech, Eilenberg, Lefschetz,
Veblen, and Vietoris) that were all called “homology theories”. It wasn’t until 1945 when
Samuel Eilenberg and Norman Steenrod gave the first (and still used) definition of what an
(ordinary) (co-)homology theory should be, based on the similarities between the different,
then known theories.
1.
The Eilenberg-Steenrod Axioms
We fix some notation here throughout the text: We denote by Top, Top(2) , Top(3) the
categories of topological spaces, pairs of spaces (called “pairs” for short), and triples of spaces
respectively. I.e. the objects of Top(2) are pairs (X, A), where X ∈ Ob(Top) is a topological
space and A ⊂ X and a morphism f : (X, A) → (Y, B) is a continuous map f : X → Y with
f A ⊂ B. Analogously the objects of Top(3) are triples (X, A, B), where X ∈ Ob(Top) and
B ⊂ A ⊂ X and a morphism f : (X, A, B) → (Y, A0 , B 0 ) is a continuous map f : X → Y with
f A ⊂ A0 and f B ⊂ B 0 . We use the term “inclusion” for maps in Top(2) or Top(3) to mean
“inclusion in each component”. If x ∈ X is a point, we will also write (X, x) for (X, {x})
(and the same for triples). Moreover, we fix the term “space” to mean “topological space”
and assume all maps to be continuous unless otherwise stated. We get canonical inclusions
Top → Top(2) → Top(3)
by sending each space X to (X, ∅) and (X, A) to (X, A, ∅) and in this way we can view Top
(resp. Top(2) ) as a full subcategory of Top(2) (resp. Top(3) ). We will use this identification
throughout the rest of the text and so, we will usually write X to mean (X, ∅).
Alternatively, one could also send X to (X, X) and (X, A) to (X, A, A). It’s not
surprising that these two types of inclusions constitute to two adjunctions. If we denote the
Chapter 1. Axiomatic Homology Theory
4
first inclusion by F and the second one by G the adjunctions are as follows:
U
Top(2) o _
/
/
F
Top
and
G
Top o _
Top(2) ,
U
where U : Top(2) → Top is the forgetful functor (X, A) 7→ X (similarly for Top(3) and
Top(2) ).
We notice that Top(2) and Top(3) are bicomplete (i.e. have small limits and colimits)
and the (co-)limits are given by taking them componentwise. For example, if we have a family
Q
Q
(Xj , Aj )j∈J of objects in Top(2) then their product is given by ( j∈J Xj , j∈J Aj ).
(1.1)
Notation. We fix the notation I := [0, 1] to denote the unit interval throughout
the whole text.
(1.2)
iff
Definition. A subcategory C 6 Top(2) is called admissible for homology theory
(i) C contains a space {∗} consisting of a single point (i.e. a final object in Top).
Furthermore, C contains all points (in Top). That means that for X ∈ Ob(C) and
∼ {∗}, we have
1=
HomC (1, X) = HomTop(2) (1, X) = HomTop (1, X).
At this point, let us fix 1 to mean a fixed one-point space in C.
(ii) If (X, A) ∈ Ob(C) then the following diagram of inclusions (called the lattice of
(X, A)) lies in C, too
(X, ∅)
:
uu
uu
u
uu
uu
/ (A, ∅)
II
II
II
II
I$
(∅, ∅)
JJ
JJ
JJ
JJ
J$
(X, A)
:
tt
tt
t
tt
tt
/ (X, X)
.
(A, A)
Moreover, we require that for f : (X, A) → (Y, B) in C, C also contains all the
maps from the lattice of (X, A) to that of (Y, B), induced by f .
(iii) For any (X, A) ∈ Ob(C), the follwoing diagram lies in C
ι0
(X, A)
ι1
/
/ (X × I, A × I)
,
where ιt : X → X × I, x 7→ (x, t) for t ∈ {0, 1}.
(1.3)
Remark. We notice that axioms (i) and (ii) imply that C really contains all points
(i.e. also points in Top(2) ). That means, for any (X, A) ∈ Ob(C) (and not only for the
(X, ∅) as in (i)) C contains all maps (1, ∅) → (X, A). The reason being that C contains the
inclusion (X, ∅) → (X, A). Moreover, it follows that C contains I since C contains 1 and
1×I ∼
= I.
Paragraph 1. The Eilenberg-Steenrod Axioms
5
(1.4)
Example. The following categories are all examples of admissible categories for
homology theory.
• Top(2) , which is the largest admissible category.
• The full subcategory of Top(2) , consisting of all pairs of compact spaces.
• The subcategory of Top(2) , having as objects all pairs (X, A), where X is locally
compact Hausdorff and A ⊂ X is closed and as arrows all maps of pairs, satisfying
that the preimage of compact subsets are compact.
(1.5)
Definition. A homotopy between two maps f0 , f1 : (X, A) → (Y, B) in C is a map
f : (X × I, A × I) → (Y, B),
in C satisfying f0 x = f (x, 0) and f1 x = f (x, 1). That means that f is an ordinary homotopy
from f0 : X → Y to f1 : X → Y , viewed as maps in Top with the additional requirement,
that f (A, t) ⊂ B ∀t ∈ I. For t ∈ I, we write ft : (X, A) → (Y, B), x 7→ f (x, t) and will loosely
refer to this family of maps as a homotopy from f0 to f1 . As always, we call f0 and f1 as
above homotopic iff there is a homotopy f in C from f0 to f1 .
With the notation from the last definition, a homotopy between f0 and f1 is a
diagram of the form
ι0
(X, A)
ι1
/
/ (X × I, A × I)
f
/ (Y, B)
,
satisfying f ◦ ι0 = f0 and f ◦ ι1 = f1 .
(1.6)
Definition. For C an admissible category, we define the so-called restriction functor
ρ : C → C which sends (X, A) to (A, ∅) and f : (X, A) → (Y, B) to ρf =: f |A
B : (A, ∅) →
(B, ∅), x 7→ f x. This functor is well-defined by axiom (ii) in the definition of an admissible
category.
(1.7)
Definition. A homology theory on an admissible category C consists of a family of
functors (Hn : C → A)n∈Z , where A is an abelian category and a family of natural transformations (∂n : Hn → Hn−1 ◦ ρ)n∈Z . Hn (X, A) is called the nth homology of (X, A) and ∂n
the nth boundary operator or connecting morphism. As mentioned before, we identify X with
(X, ∅) and in the same spirit write Hn X or Hn (X) for Hn (X, ∅), which we call the nth
(absolute) homology of X. To avoid unnecessarily complicated notation, we write f∗ for
Hn f : Hn (X, A) → Hn (Y, B) where f : (X, A) → (Y, B) and we will usually omit the index
and write ∂ for ∂n . Explicitly, ∂ being a natural transformation means that the following
diagram commutes for all f : (X, A) → (Y, B) in C.
(X, A)
f
(Y, B)
Hn (X, A)
f∗
∂
Hn (Y, B)
/ Hn−1 A
∂
f∗
/ Hn−1 B
.
These are required to satisfy
(i) (Homotopy Invariance) For each homotopy (ft )t∈I in C we have f0 ∗ = f1 ∗ .
Equivalently, with the above notation, we could also require (ι0 )∗ = (ι1 )∗ .
Chapter 1. Axiomatic Homology Theory
6
(ii) (Long Exact Homology Sequence) For each (X, A) ∈ Ob(C) we have a long
exact sequence
∂
∂
. . . → Hn+1 (X, A) −
→ Hn A → Hn X → Hn (X, A) −
→ ...,
where the unnamed arrows are induced by the canonical inclusions.
(iii) (Excision Axiom) If (X, A) ∈ Ob(C), U ⊂ X open with U ⊂ Å and the standard
inclusion (X \ U, A \ U ) → (X, A) lies in C. Then this inclusion induces for each
n ∈ Z an isomorphism
Hn (X \ U, A \ U ) ∼
= Hn (X, A),
called the excision of U .
Some authors require a weaker form of the excision axiom instead of the one before.
(iii)∗ (Weak Excision Axiom) If (X, A) ∈ Ob(C), U ⊂ X and f : X → I is a map,
satisfying U ⊂ f −1 0 ⊂ f −1 [0, 1[ ⊂ A and the inclusion (X \ U, A \ U ) → (X, A) lies
in C. Then this inclusion induces for each n ∈ Z an isomorphism
Hn (X \ U, A \ U ) ∼
= Hn (X, A),
called the excision of U .
For 1 ∈ Ob(C) a one-point space the Hn 1 are called the coefficients of the homology theory.
If furthermore the following axiom is satisfied, we speak of an ordinary homology theory.
(iv) (Dimension Axiom) If 1 ∈ Ob(C) is a one-point space then
Hn 1 = Hn (1, ∅) = 0
∀n ∈ Z \ {0}.
So in an ordinary homology theory only the coefficient H0 1 is of any interest. If we have chosen
an isomorphism H0 1 ∼
= G ∈ A we call this an ordinary homology theory with coefficients in
G and write Hn (X, A; G) := Hn (X, A).
2.
First Consequences
For the rest of this chapter, (Hn : C → A)n∈Z , (∂n )n∈Z is a given (not necessarily ordinary)
homology theory and all spaces and maps are assumed to be admissible (i.e. lie in C). As
a first remark we look at the homology of an empty space and at the homology of a space,
relative to itself (i.e. the homology of a pair (X, X)). Using the long exact homology sequence,
one easily deduces (a) in the following remark. And using the homotopy invariance axiom
(and functoriality of Hn ) one deduces the first part of (b) and with the long exact homology
sequence of (X, A) one proves the second part.
(2.1)
Remark. Let X be a topological space.
(a) Hn (X, X) = 0 ∀n ∈ Z and as a special case Hn ∅ = Hn (∅, ∅) = 0 ∀n ∈ Z.
Paragraph 2. First Consequences
7
(b) If f : A → X is a homotopy equivalence then f∗ : Hn A → Hn X is an isomorphism.
In particular if A is a deformation retract of X (i.e. the inclusion i : A ,→ X is a
homotopy equivalence) then i∗ : Hn A → Hn X is an isomorphism and Hn (X, A) =
0.
More generally, one immediately deduces the following from part (b) of the last
remark, the long exact homology sequences for (X, A) and (Y, B), and the 5-Lemma.
(2.2)
Remark. If f : (X, A) → (Y, B) is such that f : X → A and f |A
B : A → B are
homotopy equivalence then f∗ : Hn (X, A) → Hn (Y, B) is an isomorphism.
As a next step, we’re going to study the homology of a finite topological sum (i.e.
a coproduct of topological spaces). Of course, one will immediately ask questions about the
dual situation (i.e. the homology of a product) which would lead to the definition of the
so-called cross product in homology. But for now let us concentrate on the coproduct.
(2.3)
Theorem. The homology functors preserve finite coproducts. Explicitly, for pairs
(X1 , A1 ), (X2 , A2 ) let (X, A) := (X1 q X2 , A1 q A2 ) be their coproduct (i.e. topological sum)
with the standard inclusions it : (Xt , At ) → (X, A), t ∈ {1, 2}. Then for all n ∈ Z the diagram
(i1 )∗
(i2 )∗
Hn (X1 , A1 ) −−−→ Hn (X, A) ←−−− Hn (X2 , A2 )
is a coproduct in A (and so Hn (X, A) is even a biproduct since A is abelian). Put differently,
Hn (X1 , A1 ) ⊕ Hn (X2 , A2 )
(i1 )∗
(i2 )∗
/ Hn (X, A)
is an isomorphism.
(i )
Proof. Consider the morphism (i12 )∗∗ from the direct sum of the long exact homology sequences for (X1 , A1 ) and (X2 , A2 ) to the long exact homology sequence for (X, A). In view
of the 5-lemma it is enough to show the proposition for the case where A1 = A2 = ∅. We
have the standard inclusions
j1
i
1
X1 −
→
X −→ (X, X1 )
and
j2
i
2
X2 −
→
X −→ (X, X2 ),
whose induced morphisms can be combined in a commutative diagram
f1
Hn X1H
HH
HH
HH
(i1 )∗ HH$
/ Hn (X, X2 )
8
rrr
r
r
rr
rrr (j2 )∗
Hn X L
LLL
v:
L(j
LL1L)∗
LL&
/ Hn (X, X1 )
(i2 )∗ vvv
v
vv
vv
Hn X2
.
f2
By the long exact homology sequences the diagonals are exact and by the excision axiom
any morphism of the form Hn (Y, B) → Hn (Y q Z, B q Z), induced by the inclusion, is an
isomorphism. So in particular, f1 and f2 are isomorphisms. The following lemma gives the
desired result.
Chapter 1. Axiomatic Homology Theory
8
Lemma. Given a commutative diagram
(2.4)
f1
/B
u: 2
II
u
I
uu
i1 I$
uu j2
: X II j
i2 uuu
II1
II
u
$
uu
/ B1
A2
A1 I
I
f2
in an abelian category with exact diagonals. Then the following conditions are equivalent:
(a) f1 and f2 are isomorphisms;
(b)
i1
i2
: A1 ⊕ A2 → X is an isomorphism;
(c) (j1 , j2 ) : X → B1 ⊕ B2 is an isomorphism.
Considering the long exact homology sequence of a pair, one is forced to ask whether
there is an analogue for a triple (X, A, B) and indeed there is. There are topological proofs
for this but we prefer an algebraic one (even if that means that we have to draw a nasty
diagram) since it uses only the Long Exact Homology Sequence axiom.
For a triple (X, A, B) ∈ Ob(Top(3) ) with (X, A), (X, B), (A, B) ∈ Ob(C), we define
another boundary operator
∂
∂ : Hn+1 (X, A) −
→ Hn A → Hn (A, B),
where the first morphism is the boundary map given by our homology theory and the second morphism is induced by the inclusion (A, ∅) → (A, B). We shall also write ∂ for this
morphism as there should be no risk of confusion.
If we now consider the three pairs (X, A), (X, B), and (A, B), we can put their long
exact homology sequences into a so-called braid diagram
(1)
MMM
M&
H
(2)
(3)
n+1
q8
qqq
MM∂M
M&
qMM
qq∂q MMMMM
q
q
MM&
qqq
(X, A)
MMM ∂
MMM
&
qMM
qq∂q MMMMM
q
q
MM&
qqq
H (A, B)
n
qq8
q
q
qq
Hn A M
MMM
qq8
q
MM&
q
qq
MMM
MM&
Hn−1 B
8
MMM
∂ qq
MMM
q
q
q
&
H (X, B)
n
qq8
q
q
qq
qMM
qqq MMMMM
q
q
M&
qqq
MMM
MM&
q8
qqq
Hn−1 A
MMM
q8
q
M&
q
qq
∂
Hn (X, A)
Hn B M
Hn X M
∂ q8
MMM
MMM
MMM
q8
qq8
q
q8
q
MMM ∂
q
qqq
MMM
M
q
q
M
q
q
qqq
M
q
M
q
q
M
M
q
M
q
q
Mqq
Mqq
Mqq
(4)
,
where the sequences (1), (3), and (4) are the long exact homology sequences of (X, A), (X, B),
and (A, B) respectively. The sequence (2) will be called the long exact homology sequence
for the triple (X, A, B). One easily checks that this is a chain complex (i.e. the composition
of two morphisms is 0) and the following lemma gives us exactness.
(2.5)
Lemma. (Braid Lemma) If we have a braid diagram as above in any abelian
category (where the homologies are replaced by arbitrary objects of this category), where
three of the sequences are exact and the fourth is a chain complex then this will be exact,
too.
Paragraph 3. Reduced Homology
9
(2.6)
Definition. By the above the following definition makes sense. For each triple
(X, A, B) we have an exact sequence
∂
∂
. . . → Hn+1 (X, A) −
→ Hn (A, B) → Hn (X, B) → Hn (X, A) −
→ ...,
where ∂ is the boundary morphism of the triple (X, A, B) as defined above. This is called
the long exact homology sequence of the triple (X, A, B).
3.
Reduced Homology
Although the coefficients Hn 1 are important, they do not contain any geometric information
whatsoever. Because of this and for the sake of readability (so that we do not always have to
carry these one-point spaces with us while doing algebraic manipulations), we want to split
them off the homologies of our spaces. This leads to the idea of reduces homology.
(3.1)
Definition. Let X be a non-empty space and p : X → 1 the unique map to a
one-point space. We define the nth reduced homology of X as
H̃n X := ker (p∗ : Hn X → Hn 1) .
For f : X → Y , we get again an induced morphism f∗ : H̃n X → H̃n Y in the obvious way.
Like that we can extend H̃n to a homotopy invariant functor Top → A (i.e. if f is a homotopy
equivalence, then f∗ is an isomorphism).
Obviously, by definition, we can calculate the reduced homology if we have the usual
(i.e. non-reduced) homology given. One could ask whether it’s also possible to go the other
way. And indeed by some elementary algebraic facts we can. If we choose a point x ∈ X
and look at the inclusion x : 1 → X and p : X → 1, we have p ◦ x = 11 and the long exact
homology sequence for (X, x) reads as
x
∂
x
∂
∗
∗
Hn X → Hn (X, x) −
....
. . . → Hn+1 (X, x) −
→ Hn 1 −→
→ Hn−1 1 −→
Because p∗ ◦ x∗ = 1Hn 1 it follows that x∗ is a monomorphism and by exactnesss im ∂ = 0. So
we can rewrite this as a short exact sequence, which splits since p∗ is a retraction of x∗ .
0
/ Hn 1 x∗ / Hn X
GG
GG
GG
p∗
G
1Hn 1 GG
# / Hn (X, x)
/0
.
Hn 1
∼
The triangle on the left gives us an isomorphism xi∗ : H̃n X ⊕ Hn 1 −
→ Hn X, where i :
H̃n X ,→ Hn X is the standard inclusion. One plainly checks this as follows:
Hn X ∼
= ker p∗ ⊕ x∗ (Hn 1) = H̃n X ⊕ x∗ (Hn 1) ∼
= H̃n X ⊕ Hn 1.
∼
By the first isomorphism theorem j∗ |H̃n X : H̃n X −
→ Hn (X, x) is an isomorphism, where
j : X → (X, x) is the standard inclusion:
Hn (X, x) ∼
= Hn X/x∗ (Hn 1) ∼
= H̃n X ⊕ x∗ (Hn 1) /x∗ (Hn 1) ∼
= H̃n X
and finally these two together give
∼ Hn (X, x)⊕Hn 1
Hn X =
by
i
x∗
j∗ |H̃
X ×1Hn 1
n
Hn X ←−−− H̃n X⊕Hn 1 −−−−
−−−−−→ Hn (X, x)⊕Hn 1,
Chapter 1. Axiomatic Homology Theory
10
which is exactly the usual splitting condition for a short exact sequence.
A special case is when X is contractible. Then x∗ is an isomorphism (by (2.1)) and
putting this into the above short exact sequence gives us that 0 → Hn (X, x) → 0 is exact
from which one easily deduces the following theorem.
(3.2)
∼ Hn (X, x) = 0 ∀n ∈ Z.
Theorem. If X is contractible, then H̃n X =
To finish this section, we are going to introduce the analogue of the long exact
homology sequence in the reduced case. As one easily verifies, this is just a special case of
the long exact homology sequence for a triple, where the triple is of the form (X, A, x), where
x ∈ A ⊂ X is a point.
(3.3)
Theorem. (Reduced Long Exact Homology Sequence) Let (X, A) be a pair
with A 6= ∅. Then the image of the boundary operator ∂ : Hn+1 (X, A) → Hn A lies in H̃n A.
As a consequence, by restricting the long exact homology sequence of the pair (X, A), we get
another long exact sequence for the reduced homology
∂
∂
. . . → Hn+1 (X, A) −
→ H̃n A → H̃n X → Hn (X, A) −
→ ....
Proof. Consider the unique arrow p : X → 1 (resp. p : A → 1 or p : (X, A) → (1, 1)). Then
the long exact sequences of (X, A) and (1, 1) yield
...
/ Hn+1 (X, A)
...
/ Hn+1 (X, A)
p∗
...
0
∂
∂
/ H̃ X
n
/ Hn (X, A)
/ Hn A
/ Hn X
/ Hn (X, A)
p∗
/ Hn+1 (1, 1)
/ H̃ A
n
0
p∗
/ Hn 1
∼
/ ...
∂
/ ...
p∗
/ Hn 1
∂
0
/ Hn (1, 1)
0
/ ...
In the lower long exact sequence, we have used that Hn (1, 1) = 0 and get all the 0-morphisms.
Either by exactnesss or by the fact that 1 → 1 is a homeomorphism, we conclude that
Hn 1 → Hn 1 is an isomorphism. The upper row consists simply of the kernels of the corresponding vertical morphisms p∗ (observe that ker (p∗ : Hn (X, A) → Hn (1, 1)) = Hn (X, A)
since Hn (1, 1) = 0). By naturality of ∂ the lower squares in the diagram commute and so,
since p∗ ◦ ∂ = 0 ◦ p∗ we have im ∂ ⊂ ker p∗ which proves the first part of the proposition (this
actually proves more generally that the induced morphisms in the upper row are well-defined).
For the second part, we observe that all the p∗ are epimorphisms for if we choose
any point x : 1 → A (resp. x : 1 → X or x : (1, 1) → (X, A)), we have that p ◦ x = 11 and
so p has a section. By a general theorem (whose proof is left as an exercise) which says that
if we have an epimorphism of exact sequences then its kernel is also exact (and dually for a
monomorphism of exact sequences and its cokernel) we get the exactness of the reduced long
exact homology sequence.
4.
Homology of Spheres
In this section, we delve into the problem of calculating the homology of the spheres just
from the axioms. To do so, we observe first, that we can divide the sphere S n ⊂ Rn+1 in an
upper and lower hemisphere
n
D±
:= {(x1 , . . . , xn+1 ) ∈ S n | ±xn+1 > 0} .
Paragraph 4. Homology of Spheres
11
n ∼ D n by simply projecting D n along the x
n
n+1 .
Obviously D±
=
n+1 -axis to R × {0} ⊂ R
±
Moreover, we observe that we have for any n ∈ N an inclusion
S n−1 → S n
(x1 , . . . , xn ) 7→ (x1 , . . . , xn , 0)
or a little more geometric, by viewing S n−1 as the equator of S n . By combining these we get
S 0 ,→ S 1 ,→ S 2 ,→ . . . . Let’s furthermore fix the notation ei ∈ Rn to denote the ith standard
basis vector having (ei )j = δi,j (the Kronecker delta) and with this, let’s write N := en+1
and S := −en+1 for the north and south pole of S n respectively. Now, for n ∈ N>0 we look
at the commutative diagram
n , S n−1 )
Hk (D−
/ Hk (S n , D n )
/ Hk (S n , S n \ {S})
+
n , D n \ {S})
Hk (D−
−
n ,→ S n \ {S} are homotopy
induced by inclusions. Because S n−1 ,→ Dn \ {S} and D+
equivalences, it follows that the vertical arrows are isomorphisms (by remark (2.2)). For the
n and deduce that this is also an
bottom arrow, we can use the excision axiom with U := D˚+
isomorphism. In conclusion, the top arrow has to be an isomorphism, too.
We choose ∗ := e1 = (1, 0, . . . , 0) ∈ S n−1 ⊂ S n and insert this isomorphism in a
second diagram
∂
n , S n−1 )
Hk (D−
o
/ Hk−1 (S n−1 , ∗)
∼
=
H̃k−1 S n−1
σ+
n) o
Hk (S n , D+
j
Hk (S n , ∗)
∼
=
σ+
H̃k S n
n , S n−1 ) reads as
The reduced long exact homology sequence of (D−
∂
n
n
n
. . . → H̃k (D−
) → Hk (D−
, S n−1 ) −
→ H̃k−1 (S n−1 ) → H̃k−1 (D−
) → ....
n ∼ D n−1 is contractible, we deduce from theorem (3.2) that H̃ (D n ) = H̃
n
Since D−
=
k
k−1 (D− ) =
−
n is
0 and so ∂ is an isomorphism. By the same argument, j is an isomorphism, since {∗} ,→ D−
a homotopy equivalence. Now we can easily define σ+ to be the unique isomorphism making
the diagram commute.
(4.1)
Lemma. H̃n S 0 ∼
= Hn 1 for all n ∈ Z.
Proof. Choose a point x : 1 → S 0 and denote the other point by y : 1 → S 0 . Let’s also
write i : H̃n S 0 ,→ Hn S 0 for the standard inclusion. As seen in the section about the reduced
homology and theorem (2.3), we get a commutative diagram
H̃n S 0 ⊕
O Hn 1
i
x∗
∼
( y∗ )
/ H S 0 o x∗
n
∼
Hn 1 ⊕O Hn 1
O
O
Hn 1
Hn 1
,
Chapter 1. Axiomatic Homology Theory
12
where the vertical arrows are the standard
inclusions into the second summand. If we define
∼
→ Hn 1 ⊕ Hn 1, we can rewrite this as a
the isomorphism f := ( xy∗∗ )−1 ◦ xi∗ : H̃n S 0 ⊕ Hn 1 −
commutative diagram
/ Hn 1 /
0
/ H̃ S 0 ⊕ H 1
n
n
/ / H̃ S 0
n
f o
fˆ
o
/ Hn 1 /
0
/ Hn 1 ⊕ Hn 1
/0
/ / Hn 1
/0
,
where the rows are exact and fˆ is the unique arrow between the cokernels, induced by f ,
making the diagram commute. By the 3-lemma (which is the special case of the 5-lemma for
short exact sequences) it follows that fˆ is an isomorphism.
(4.2)
Theorem. For all k ∈ Z and n ∈ N we have isomorphisms
H̃k S n ∼
= Hk−n 1
and
Hk S n ∼
= Hk−n 1 ⊕ Hk 1.
It follows that we also have isomorphisms
∼ Hk−(n+1) 1.
Hk (Dn+1 , S n ) ∼
= H̃k−1 S n =
Proof. As seen in the last paragraph, we have isomorphisms
H̃k S n ∼
= H̃k−1 S n−1 ∼
= ... ∼
= H̃k−n S 0 ∼
= Hk−n 1,
where we used the above lemma for the last isomorphism. By remembering ourselves that
Hk S n ∼
= H̃k S n ⊕ Hk 1 the first part of the proposition follows. For the second part, let’s look
at the reduced long exact homology sequence for (Dn+1 , S n ). Because Dn+1 is contractible,
this looks like
∂
0 → Hk (Dn+1 , S n ) −
→ H̃k−1 S n → 0
and so ∂ is an isomorphism.
(4.3)
Corollary. Let (Hn )n∈Z , (∂n )n∈Z be an ordinary homology theory, having coefficient H0 1 ∼
= G then for n ∈ N>0
Hk S ∼
=
n
(
G k ∈ {0, n}
0 otherwise
and similarly
Hk (D
n+1
,S ) ∼
=
n
(
G k =n+1
0 otherwise .
By noticing that an ordinary homology theory with non-trivial coefficient exists (e.g.
singular homology, which will be treated in the next chapter), we easily deduce
(4.4)
Corollary. (Invariance of Dimension) Rm ∼
= Rn ⇔ m = n for m, n ∈ N.
Proof. The direction “⇐” is trivial and for the other direction, we assume that for m 6= n we
we have a homeomorphism Rm → Rn . The case where m = 0 or n = 0 is trivial an so the
case m, n > 1 is left. We can extend our homeomorphism Rm → Rn to a homeomorphism of
the one-point compactifications, which is S m ∼
= S n but since m 6= n by the above theorem
m
n
∼
Hn S = 0 but Hn S = G 6= 0.
(4.5)
Corollary. S n is not contractible ∀n ∈ N
Chapter 2
ACYCLIC MODELS
In der Algebra gibt es viele Definitionen; manche werden auch
gebraucht.
ARMIN LEUTBECHER
In this chapter we will be concerned with studying so-called acyclic models and will
prove a form of the famous acyclic model theorem. The theory of acyclic models is in some
sense a way to abstract the standard models arising in homology theory, like the standard
simplices in singular homology (see the next chapter). The acyclic model theorem will be
useful in this text to prove homotopy invariance and the excision axiom for singular homology
but generally finds wide applications throughout algebraic topology and homological algebra.
1.
Models
Let C be a category. A specified set M ⊂ Ob(C) of objects in C will be called models of C.
From now on we fix the notation M to denote a set of models of a category.
(1.1)
Example. The intuition (and our primary use for that matter) is the following:
There is a purely combinatorial theory of simplices known as simplicial sets. The easiest
“models” of this theory in a topological context are the standard simplices ∆q ⊂ Rq+1 . And
in fact, we will investigate {∆q | q ∈ N} as models of Top in the next chapter using the tool(s)
we are going to develop in this one.
(1.2)
Definition. A functor F : C → R-Mod for R any ring will be called free with
models M iff there is a subset M0 ⊂ M and for each M ∈ M0 an element eM ∈ F M such
that for every C ∈ Ob(C) the module F C is free and the set
(F a)eM ∈ F C M ∈ M0 , a ∈ C(M, C)
forms a basis for F C. Put differently, the functor F factors as
Sets
? ???

??


?

/
C
R-Mod
,
F
where Sets → R-Mod is the free construction and the functor C → Sets maps C ∈ Ob(C)
to the set discribed above and b : C → D in C to b∗ with b∗ ((F a)eM ) := F (ba)eM . If F does
map to Ch(R-Mod) (the category of chain maps between chain complexes of R-modules) we
call F free with models M iff it is free with models M at each degree Fp , where for p ∈ Z
Fp : C → R-Mod maps a : C → D to (F a)p : (F C)p → (F D)p . One should notice that at
Chapter 2. Acyclic Models
14
each degree, we can have a different subset Mp ⊂ M and different elements epM ∈ Fn M for
M ∈ Mp .
Finally, let’s call a functor F : C → Ch(R-Mod) aacyclic on the models M (aacyclic
stands for almost acyclic) iff for each M ∈ M the chain complex F M is exact everywhere
except at the degree 0. I.e. Hi (F M ) = 0 ∀i ∈ Z \ {0}. In the same spirit, we call a chain
complex aacyclic iff it is acyclic except at degree 0.
(1.3)
Definition. Let F, G : C → Ch(R-Mod) be two functors and α, β : F → G two
natural transformations. We say that α and β are naturally chain homotopic or simply
nautrally homotopic iff all their components are chain homotopic in a natural way. That is
for each C ∈ Ob(C) there is a chain map χC : F C → GC of degree 1 (i.e. (χC )n : Fn C →
Gn+1 C ∀n ∈ Z) such that
αC − βC = ∂GC ◦ χC + χC ◦ ∂F C ,
where ∂F C and ∂GC denote the boundaries of F C and GC respectively. Furthermore, χC is
required to be natural in C. That is for each a : C → D in C the following diagram commutes
C
a
χC
FC
Fa
D
FD
/ GC
χD
Ga
/ GD
.
So in some sense χ is a natural transformation F → G, which is not completely honest since
the components χC are not really arrows in Ch(R-Mod). To be even more formal, one could
say that χ is a natural transformation F → S − ◦ G where S − : Ch(R-Mod) → Ch(R-Mod)
is the shift functor that shifts a chain complex X by −1. So X is mapped to X 0 = S − X,
having Xn0 = Xn+1 with the obvious boundaries (the arrow function of S − is obvious, too).
2.
The Acyclic Model Theorem
In this section we will state and prove a form of the acyclic model theorem.
(2.1)
Theorem. (Acyclic Model Theorem) Let C be a category with models M and
F, G : C → Ch(R-Mod) functors which are 0 in negative degrees (i.e. Fn = Gn = 0 ∀n ∈
Z<0 ). If F is free with models M and G is aacyclic on M and there is a natural transformation
ϕ : H0 F → H0 G (where H0 F, H0 G : C → R-Mod) then there is a natural transformation
ϕ̃ : F → G, which induces ϕ. Moreover, ϕ̃ is unique up to natural homotopy.
(2.2)
Remark. For the sake of readability we will omit unnecessary indices in the following proof. We will assume that the attentive reader will still be capable of following it and
fill in the details.
This proof seems a little complicated at first glance (which it really isn’t). Because
of this we will briefly sketch it before formalizing it rigorously. For C ∈ Ob(C), we want to
define ϕ̃ as to make the diagram
...
∂
/ Fn+1 C
ϕ̃
...
∂
∂
∂
/ ...
∂
ϕ̃
/ Gn+1 C
/ Fn C
/ Gn C
∂
/ ...
∂
/ / H0 (F C)
/0
ϕ
ϕ̃
∂
/ F0 C
/ G0 C
/ / H0 (GC)
/0
Paragraph 2. The Acyclic Model Theorem
15
commute. We do this inductively in each degree n. To do so, we first consider the case where
C = M is a model and so the lower row is exact. We can use this exactness to “lift” ϕ̃ from
degree n to n + 1. Afterwards we use the fact that F is free with models M to extend this
to arbitrary C ∈ Ob(C).
Proof. By presumption, for each n ∈ N there is a collection Mn ⊂ M and for each M ∈ Mn
an element enM ∈ Fn M as in the definition of a free functor with models M. Now, let
C ∈ Ob(C) be arbitrary. Again by presumption, F0 C = Z0 (F C) and G0 C = Z0 (GC) are
the cycles of F C and GC at degree 0 respectively. Thus, we get standard projections onto
the 0th homology modules as in the following diagram
/ / H0 (F C)
F0 C
ϕ
ϕ̃
/ / H0 (GC)
G0 C
.
Thus, we can augment F and G by (re)defining F−1 C := H0 (F C) and G−1 C := H0 (GC)
and defining the standard projections as the boundary morphisms. By this, ϕ̃ is defined in
degree −1, where it is simply ϕ.
For the inductive step let’s assume that ϕ̃ is defined in degree n − 1 for n > 0. For
each model M ∈ Mn we consider ϕ̃(∂enM ) ∈ Gn−1 M (which we have already defined). Since
the diagram
∂
Fn M
ϕ̃
/ Fn−1 M
ϕ̃
ϕ̃
/ Gn−1 M
∂
Gn M
/ Fn−2 M
∂
/ Gn−2 M
∂
.
commutes and the lower row is exact (because G is aacyclic on M) we conclude that
∂ ϕ̃(∂enM ) = ϕ̃(∂∂enM ) = 0 and so ϕ̃(∂enM ) ∈ Gn−1 M must be a boundary. I.e. we can
choose c ∈ Gn M satisfying ∂c = ϕ̃(∂enM ) and define ϕ̃enM := c, which makes the above
diagram commute.
For a : M → C a morphism in C (Fn a)enM ∈ Fn C is a basis element of Fn C and we
define ϕ̃ ((Fn a)enM ) := (Gn a)ϕ̃enM . We do this for every a and every M and like that define ϕ̃
on the basis elements of Fn C which means that we can extend it uniquely to ϕ̃ : Fn C → Gn C.
To check that ϕ̃ thus defined is a chain morphism (i.e. commutes with the boundaries), we
look at the following cubical diagram
ϕ̃
Fn M
rr
rrr
r
r
r
y
Fn C
ϕ̃
/ Gn C
Fn−1 M
rr
rrFrn−1 a
r
r
yr
Fn−1 C
ϕ̃
/ Gn M
Gn a rrr
Fn a
ϕ̃
rr
yrrr
/ Gn−1 M
rrr
rrG
r
n−1 a
r
yr
/ Gn−1 C
,
where the downward arrows are all boundary morphisms. The left and right faces of the
cube obviously commute and the bottom commutes by inductive hypothesis. For the element
enM ∈ Fn M the top and the back faces commute by definition of ϕ̃. So the front face has
16
Chapter 2. Acyclic Models
to commute, too for (Fn a)enM ∈ Fn C. Since this holds for all M and all a the front face
commutes for all basis elements of Fn C and so commutes as a whole. This defines ϕ̃ in
degree n. One easily checks the naturality of ϕ̃, i.e. for a : C → D an arrow in C the
equation ϕ̃ ◦ F a = Ga ◦ ϕ̃ holds.
What is left to prove is the uniqueness up to natural homotopy. So suppose that
ϕ̃, ψ̃ : F → G are two natural transformations inducing ϕ in the 0th homology or put differently, that lift ϕ̃ (defined in degree −1). For each object C ∈ Ob(C) we must define a chain
homotopy χ : F C → GC from ϕ̃ to ψ̃ (i.e. χ is of degree 1 and satisfies ∂χ + χ∂ = ϕ̃ − ψ̃),
which is natural in C. χ is already defined in degree −1 (note that we are still working with
the augmented F and G), where it is simply 0 because there ϕ̃ = ψ̃ = ϕ. Suppose now, that
χ is defined in degree n − 1 with n > 0. Fn C has {(F a)enM }M,a as a basis and we notice that
b := ϕ̃enM − ψ̃enM − χ(∂enM ) is a cycle because
∂ ϕ̃enM − ∂ ψ̃enM − ∂χ∂(enM ) = ϕ̃(∂enM ) − ψ̃(∂enM ) − −χ(∂∂enM ) + ϕ̃(∂enM ) − ψ̃(∂enM ) = 0.
Because GM is aacyclic, b must be a boundary, i.e. there is a c ∈ Gn+1 M with ∂c = b and
we define χenM := c. By defining χ ((Fn a)enM ) := (Gn+1 a)χenM we have defined χ on all basis
elements and thence can extend it uniquely to χ : Fn C → Gn+1 C. One can easily check that
χ thus defined is really a chain morphism of degree 1 by using a cubical diagram, similar to
the one above. By construction this gives us a chain homotopy χ : ϕ̃ ' ψ̃ and it is plain to
check naturality in C.
(2.3)
Corollary. If F, G : C → Ch(R-Mod) are functors which are 0 in negative degrees
and both free and aacyclic on M and there is a natural isomorphism ϕ : H0 F ∼
= H0 G, then
ϕ can be extended to a natural isomorphism ϕ̃ : HF ∼
= HG, where H : Ch(R-Mod) →
Ch(R-Mod) is the homology functor.
(2.4)
Corollary. If F : C → Ch(R-Mod) is 0 in negative degrees and both free with
models M and aacyclic on M and α : F → F is a natural endotransformation inducing
the identity in 0th homology. Then α is naturally homotopic to 1F . In particular, for each
C ∈ Ob(C) there is a chain homotopy αC ' 1F C (i.e. αC and 1F C are chain homotopic).
Proof. By the acyclic model theorem there is a natural transformation ϕ : F → F which
induces 1H0 F : H0 F → H0 F and is unique up to homotopy. But α and 1F are two such
natural transformations and so α and 1F are naturally homotopic.
Chapter 3
SINGULAR HOMOLOGY
Despite physicists, proof is essential in mathematics.
SAUNDERS MAC LANE
In this chapter we are going to quickly repeat the definition of singular homology,
mainly to introduce the reader to the notation used in this text. Afterwards we are going to
prove the axioms for an ordinary homology theory in the case of singular homology.
1.
Definitions
We repeat the definition of singular homology in this section. We do so, mainly, to fix the
notation but we will also prove as a first result that the singular homology of contractible
spaces vanishes in positive degrees.
(1.1)
Definition. For p ∈ N let ei ∈ Rp+1 be the i-th standard basis vector. We define
the p-dimensional standard simplex as
n
o
∆p := (t0 , . . . , tp ) = t0 e0 + . . . + tp ep ∈ Rp+1 t0 + . . . + tp = 1 and ti > 0 ∀i .
Every function α : {0, . . . , p} → {0, . . . , q} induces an affine map
p
q
∆α : ∆ → ∆ ,
p
X
ti ei 7→
i=0
p
X
ti eαi .
i=0
Especially interesting for our theory is the injective map δip : {0, . . . , p − 1} → {0, . . . , p} for
i ∈ {0, . . . , p} which simply leaves out the value i. We define dpi := ∆δip and will usually
omit the upper index where it is unnecessary. For X a topological space, we call a map
σ : ∆p → X a singular p-simplex in X and call σ ◦ dpi its ith face. Furthermore, for each
p ∈ Z we define a functor Sp : Top → AbGrp, which is 0 for p < 0 and otherwise sends
a space X to the free abelian group over all singular simplices σ : ∆p → X and a map
f : X → Y to f∗ : Sp X → Sp Y which is defined on the basis elements of Sp X by f∗ σ := f ◦ σ.
An element of Sp X is called a singular p-chain. The faces of a singular simplex σ : ∆p → X
can be used to define a morphism
∂p : Sp X → Sp−1 X, σ 7→
p
X
(−1)i σ ◦ dpi
i=0
and again, we omit the indices where they are clear from the context. We call ∂p the pth
boundary morphism or simply the pth boundary. One easily checks that
p
δjp+1 ◦ δip = δip+1 ◦ δj−1
∀p ∈ N, i ∈ {0, . . . , p}, j ∈ {0, . . . , p + 1}, i < j
Chapter 3. Singular Homology
18
and concludes that ∂p ◦ ∂p+1 = 0 ∀p ∈ N. Thus the Sp can be put together to yield a functor
S : Top → Ch(AbGrp),
(where Ch(AbGrp) is the category of chain maps between chain complexes of abelian groups)
∂
∂
∂
by sending a space X to the chain complex . . . −
→ Sp+1 X −
→ Sp X −
→ . . . and a map f : X → Y
to f∗ : Sp X → Sp Y in each degree. We write Hp X := Hp (SX) and call this the pth
singular homology of X (with coefficients in Z). As is well known from a basic course about
homological algebra f∗ : SX → SY induces a morphism f∗ : HX → HY and this gives us
a functor H : Top → Ch(AbGrp) (recall that the Hp (SX) form a chain complex with all
boundary morphisms being 0). This gives us the absolute homology of a space X but to be
a full-fledged homology theory, we must define relative homology, too. So if i : A ,→ X is an
inclusion, we define S(X, A) := SX/i(SA) = SX/SA = coker(i∗ : SA ,→ SX), which defines
a functor S : Top(2) → Ch(AbGrp) again by sending f : (X, A) → (Y, B) to the morphism
f∗ : SX/SA → SY /SB induced by f∗ : SX → SY , the former being well-defined as f A ⊂ B.
As usual, we call Hp (X, A) := Hp (S(X, A)) the pth singular homology of X relative to A
and again get a functor H : Top(2) → Ch(AbGrp).
So, by definition, for (X, A) ∈ Ob(Top(2) ) we get a short exact sequence of chain
complexes SA / / SX / / S(X, A) induced by the standard inclusions and again by elementary homological algebra, we can already deduce one of the Eilenberg-Steenrod axioms.
(1.2)
Theorem. (Long Exact Homology Sequence) For (X, A) ∈ Ob(Top(2) ) the
standard inclusions A ,→ X ,→ (X, A) yield a long exact sequence
∂
∂
... −
→ Hp A → Hp X → Hp (X, A) −
→ Hp−1 A → . . . → H0 (X, A) → 0.
To prove the homotopy invariance and excision axiom for the singular theory, it will
be useful to consider a special case of the former axiom first.
(1.3)
Theorem. If X is a contractible space. Then SX is aacyclic (i.e. Hp X = 0 ∀p 6= 0).
Furthermore, H0 X is generated by one element.
Proof. Let h : X × I → X be a homotopy 1X ∼
= x0 , where x0 : X → X, x 7→ x0 is a constant
map. We define a chain homotopy D : 1SX ' 0, i.e. a chain map of degree 1 satisfying
∂D + D∂ = 1SX . To do so, we consider a singular simplex σ : ∆p → X (i.e. a basis element
of Sp X) and define Dσ : ∆p+1 → X by
( Dσ(t0 , . . . , tp ) =
h σ
x0
tp
t1
1−t0 , . . . , 1−t0
, t0
t0 =
6 1
t0 = 1
(notice that t0 + . . . + tp = 1 and so 1 − t0 = t1 + . . . + tp ). The faces of Dσ satisfy
(Dσ)di = D(σdi−1 ) for i > 0 and (Dσ)d0 = σ. So for p > 1 we calculate
∂Dσ = (Dσ)d0 −
p+1
X
i−1
(−1)
i=1
(Dσ)di = σ −
p
X
(−1)i D(σdi ) = σ − D∂σ.
i=0
For p = 0 we get ∂Dσ − D∂σ = ∂Dσ = σ − σ0 , where σ0 : ∆0 → X, 1 7→ x0 . From this
we conclude that D is really a chain homotopy as required and so Hp X = 0 ∀p 6= 0 and for
p = 0 our simplices σ and σ0 differ by a boundary, meaning that [σ] = [σ0 ] ∈ H0 X and so
H0 X is an abelian group with one generator.
Paragraph 2. Homotopy Invariance
2.
19
Homotopy Invariance
In this section, we are going to prove the first of Eilenberg and Steenrod’s axioms for our
singular theory. So we must prove that for X a topological space ι0 (X) and ι1 (X) induce
the same morphisms in homology, where ιt (X) : X → X × I, x 7→ (x, t) for t ∈ {0, 1}. As
promised in chapter 2, we are going to use the following models to apply the acyclic model
theorem to singular homology.
(2.1)
Definition. For the rest of this chapter, we will write S for the collection
S := {∆q | q ∈ N}
of objects of Top and call these the standard models. They will be key in applying the acyclic
model theorem to singular homology.
(2.2)
Remark. The functor S : Top → AbGrp is free with models S and aacyclic on
S. It is obviously aacyclic since all the ∆q are contractible. The freeness has to be checked
at each degree. To do so, let p ∈ N and we consider S 0 := {∆p } ⊂ S and the element
ep := 1∆p ∈ Sp ∆p . By definition for X any space, the set {σ∗ ep = σ ∈ Sp X | σ : ∆p → X}
forms a basis for Sp X.
(2.3)
Theorem. (Homotopy Invariance) For X a topological space and ι0 (X), ι1 (X)
as above the induced morphisms (ι0 (X))∗ , (ι1 (X))∗ : SX → S(X × I) are chain homotopic
(even natural in X) and so they induce the same morphism HX → H(X × I) in homology.
Proof. Consider the functor G : Top → Ch(AbGrp) mapping X to S(X ×I) and f : X → Y
to (f × 1I )∗ : S(X × I) → S(Y × I). Since M × I is contractible for all M ∈ S this functor
is aacyclic on S (cf. (1.3)). For F := S : Top → Ch(AbGrp), which is free with models
S as seen above, we can apply the acyclic model theorem. Clearly ι0 , ι1 : F → G are
two natural transformations and they induce the same morphism H0 X → H0 (X × I). To
see this, let σ : ∆0 → X, 1 7→ x0 be an arbitrary 0-simplex (i.e. a point) and consider
b := (ι1 (X))∗ σ − (ι0 (X))∗ σ ∈ S0 (X × I). Define c : ∆1 ∼
= I → X × I, t 7→ (x0 , t) and it
follows that ∂c = (x0 , 1) − (x0 , 0) and so ∂c = ι1 (X) ◦ σ − ι0 (X) ◦ σ meaning that (ι0 (X))∗ σ
and (ι1 (X))∗ σ differ only by a boundary and so are the same in homology.
3.
Barycentric Subdivision
(3.1)
Definition. Let v0 , . . . , vp ∈ ∆q . We define [v0 , . . . , vp ] to be the affine singular
simplex
p
q
[v0 , . . . , vp ] : ∆ → ∆ ,
p
X
i=0
ti ei 7→
p
X
ti vi
i=0
and dub ξ p := [e0 , . . . , ep ] = 1∆p : ∆p → ∆p . One easily calculates that for p > 1
∂[v0 , . . . , vp ] =
p
X
(−1)i [v0 , . . . , vˆi , . . . , vp ].
i=0
Geometrically, [v0 , . . . , vp ](∆p ) ⊂ ∆q ⊂ Rq+1 is the (possibly degenerate) p-simplex spanned
by v0 , . . . , vp . For v ∈ ∆q and p > 0, we define the v-cone over σ = [v0 , . . . , vp ] as
vσ = v[v0 , . . . , vp ] := [v, v0 , . . . , vp ].
Chapter 3. Singular Homology
20
Geometrically, this is just a new simplex constructed from σ by adding another point v “over”
σ and taking the convex hull. We now extend the definition of the v-cone linearly to p-chains
c ∈ Sp ∆q consisting of affine singular simplices. Explicitly, if σj = [v0j , . . . , vpj ], p > 0 is an
P
affine singular simplex and mj ∈ Z for j ∈ {1, . . . , n}, we define for c = nj=1 mj σj ∈ Sp ∆q
Pn
the v-cone vc := j=1 mj vσj ∈ Sq+1 ∆p . As always for homomorphisms, we define v0 = 0
and so in particular v : Sp ∆q → Sp+1 ∆q is the zero morphism for p < 0. This is not a
wholely trivial remark as some people put [ ] := 0 by applying the above law for calculating
the boundary in the case p = 0 (so one has ∂[v0 ] = [ ] = 0) but then one is tempted to
calculate v[ ] = [v], which cannot be the case since [ ] = 0 and v is a homomorphism. To avoid
such unnecessary confusion, we leave the expression [ ] undefined and will need an additional
case differentiation here and there.
(3.2)
Lemma. For v ∈ ∆q and σ = [v0 , . . . , vp ] : ∆p → ∆q an affine singular simplex we
have ∂vσ = σ − v∂σ for p > 1 and ∂vσ = σ − [v] for p = 0.
Proof. For p > 1 one easily calculates
∂vσ = ∂[v, v0 , . . . , vp ] = [v0 , . . . , vp ] −
p
X
(−1)i [v, v0 , . . . , vˆi , . . . , vp ] = σ − v∂σ
i=0
and similarly for p = 0
∂vσ = ∂[v, v0 ] = [v0 ] − [v] = σ − [v].
An especially interesting case of a v-cone is the case when v is the barycentre of
σ and we look at the v-cone over the faces of σ. This will lead to the so-called barycentric
subdivision. Let σ = [v0 , . . . , vp ] : ∆p → ∆q be an affine singular simplex. We write bσ :=
Pp
Pp
1
1/p+1
i=0 ei .
i=0 vi for the barycentre of σ and as a special case, we write bp := bξ p = /p+1
We can now define the barycentric subdivision B recursively over p. To do this for an
arbitrary singular simplex, we must first define the barycentric subdivision for the universal
simplices ξ p
The geometric idea of barycentric subdivision is to divide a p-simplex into smaller
and smaller simplices so that at some point these simplices are small enough as to fit entirely
into an open subset of a given cover (i.e. their diameter is smaller than a Lebesgue number
of this cover) thus enabling us to use Lebesgue’s Lemma. For p = 0 ξ p is just a point and
we cannot subdivide anything. For p = 1 the (image of the) simplex ξ p is an interval and we
identify this with [0, 1]. We subdivide it in the two pieces [0, 1/2] and [1/2, 1], i.e. we connect
the endpoints to the barycentre. Assume now, that the barycentric subdivision is defined for
dimension p − 1 then we define it for ξ p by subdividing each face (which are p − 1-simplices)
and taking the bp -cone, i.e. connecting each face with the barycentre. Pictorially (here for
the standard 2-simplex ξ 2 ) this is just
4
444
44
44
44
4
/o /o /o /
4
444
4
O
OOOooo444
oO
ooo OOOO444
O
o
oo
/o /o /o /
4((4
,,
O((4 44
oo O( O
oO
ozOO,,Ooo(DoOD444
ddoZ?OZZ
zd
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z

g
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(3.3)
Definition. (Barycentric Subdivision) For X a topological space and c ∈ S0 X
a 0-chain we define the barycentric subdivision of c as BX c := c. Assume now that BX is
defined for all topological spaces X and all (p − 1)-chains in Sp−1 X (with p > 1). To extend
this to p-chains, we first define the barycentric subdivision B∆p ξ p of the universal p-simplex
Paragraph 3. Barycentric Subdivision
21
ξ p = 1∆p by B∆p ξ p := bp B∆p (∂ξ p ) (i.e. subdivide the faces and connect with the barycentre).
For an arbitrary topological space X we define BX : Sp X → Sp X on the basis elements (i.e.
the singular p-simplices) and extend this linearly to all p-chains. So if σ : ∆p → X is any
singular p-simplex we subdivide σ by writing σ = σ ◦ ξ p and subdividing ξ p . I.e. we define
BX σ := σ∗ B∆p ξ p .
For the following theorem, in the case where X = ∆q for some q ∈ N, let us fix the
notation Ap X to denote the subgroup of Sp X spanned by the affine singular simplices. An
element of Ap X will be called an affine p-chain. By (3.2) the boundary maps ∂ map Ap+1 X
to Ap X, i.e. the image of an affine p + 1-chain is an affine p-chain. So the Ap X together
with the singular chain maps form again a chain complex AX. Obviously, if f : X → X 0
0
is continuous and X 0 = ∆q for some q 0 ∈ N then f∗ : SX → SX 0 restricts to a chain
morphism f∗ : AX → AX 0 and in particular the barycentric subdivision restricts to a map
BX : Ap X → Ap X by definition of BX .
(3.4)
Theorem. For X a topological space BX : SX → SX is a natural chain map. Put
differently, B is a natural endotransformation B : S → S. Explicitly this means
(a) BX : Sp X → Sp X is natural in X for each p ∈ Z. I.e. for f : X → Y in Top we
have f∗ ◦ BX = BY ◦ f∗ .
(b) If X = ∆q for some q ∈ N and σ = [v0 , . . . , vp ] : ∆p → X is an affine singular
simplex then BX σ = bσ BX ∂σ.
(c) BX is compatible with the boundary morphisms. I.e. BX ◦ ∂ = ∂ ◦ BX .
Proof. Ad (a): This is trivial. Actually, we really defined the barycentric subdivision by
naturality. One only needs to remember the functoriality of S. To be precise (f ◦g)∗ = f∗ ◦g∗ .
We can now check the naturality of B on a basis element σ : ∆p → X in Sp X.
f∗ BX σ = f∗ σ∗ (bp B∆p ∂ξ p )
and also
BY f∗ σ = (f ◦ σ)∗ (bp B∆p ∂ξ p ) .
Ad (b): This follows easily from (a):
(a)
BX σ = σ∗ B∆p ξ p = σ∗ (bp B∆p ∂ξ p ) = bσ σ∗ B∆p ∂ξ p = bσ BX σ∗ ∂ξ p = bσ BX ∂σ.
Ad (c): We first prove our proposition for X = ∆q and affine singular simplices by induction
on p. The case p = 0 is trivial as BX : S0 X → S0 X is simply the identity map. For p = 1
let σ : ∆1 → X be an affine singular 1-simplex. Then ∂σ is a 0-chain and so BX ∂σ = ∂σ by
definition. On the other hand
∂BX σ = ∂(bσ ∂σ) = ∂ ([bσ , σe1 ] − [bσ , σe0 ]) = [σe1 ] − [bσ ] − ([σe0 ] − [bσ ]) = ∂σ.
For p > 2 we can apply (3.2) and calculate for an affine singular simplex σ : ∆p → X
(3.2)
∂BX σ = ∂(bσ BX ∂σ) = BX ∂σ − bσ ∂BX ∂σ
ind. hyp.
=
BX ∂σ − bσ BX ∂∂σ = BX ∂σ.
We conclude that for X = ∆q we have BX ◦ ∂ = ∂ ◦ BX on AX. If X is now an arbitrary
space and σ : ∆p → X a singular simplex, we have
∂BX σ = ∂σ∗ B∆p ξ p = σ∗ ∂B∆p ξ p = σ∗ B∆p ∂ξ p = BX σ∗ ∂ξ p = BX ∂σ,
where we used in the third equality that ∂ ◦ B∆p = B∆p ◦ ∂ on A∆p as shown above.
Chapter 3. Singular Homology
22
(3.5)
Remark. If our topological space X is fixed and there is no risk of confusion, we
will usually omit the index and simply write B for BX .
(3.6)
Remark. Each simplex in the p-chain B∆q σ is a subsimplex of σ (i.e. the image of
these simplices in ∆q is contained in σ(∆p )). This follows directly from the definition of B∆q .
As was mentioned above, we want to use the barycentric subdivision to apply
Lebesgue’s Lemma. I.e. for a given open cover we want to apply the barycentric subdivision until the diameter of our simplices is smaller than a Lebesgue number of the cover.
To do so, we have to investigate how the diameter of the simplices considered changes under
barycentric subdivision.
Theorem. Let σ = [v0 , . . . , vp ] : ∆p → ∆q be an affine singular simplex. Then
(3.7)
(a) diam σ(∆p ) = max {kvi − vj k | i, j ∈ {0, . . . , p}};
(b) For all x ∈ σ(∆p ) kbσ − xk 6
p
p+1
diam σ(∆p );
(c) Every simplex in the chain Bσ = B∆q σ has diameter at most
p
p+1
diam σ∆p .
Proof. Ad (a): Clearly maxi,j kvi − vj k 6 diam σ(∆p ) since vi ∈ σ(∆p ) ∀i. So we have to
P
P
prove the other inequality. For x ∈ Rq+1 and y = pi=0 ti vi ∈ σ(∆p ) with pi=0 ti = 1 and
ti > 0 ∀i we calculate
p
p
p
p
X
X
X
X
kx − yk = ti x −
t i vi 6
ti kx − vi k 6
ti max kx − vj k = max kx − vj k .
j
j
i=0
i=0
i=0
i=0
Pp
Putting x = i=0 si vi ∈ σ(∆p ), we get the desired result by a similar calculation.
Pp
Pp
1 Pp
p
Ad (b): By definition bσ = p+1
j=0 tj = 1 and
j=0 tj vj ∈ σ(∆ ) with
i=0 vi . If x =
tj > 0 ∀j it is easy to see that for all j ∈ {0, . . . , p}
p
p
p
1 X
1 X
p+1 1 X
kbσ − vj k = vi − vj = vi −
vj = (vi − vj )
p + 1
p + 1
p + 1
p
+
1
i=0
i=0
i=0
p
p
p
1 1 X
1 X
X
=
kvi − vj k =
kvi − vj k
(vi − vj ) 6
p+1
p+1
p + 1 i=0
i=0
i=0
i6=j
p
6
max kvi − vj k .
p+1 i
And so
kbσ − xk 6
p
X
j=0
p
tj kbσ − vj k 6
p X
p
tj max kvi − vj k =
diam σ(∆p ).
i,j
p + 1 j=0
p+1
Ad (c): For this proof, for the sake of readability, let us fix the notation mesh c for a pP
chain c = m
i=0 ni σi (with ni 6= 0 ∀i) to denote the maximum diameter of the σi . I.e.
mesh c := max {diam σi (∆p ) | i ∈ {0, . . . , m}}. Now the proof goes by induction on p. For
p = 0, σ(∆p ) ∈ ∆q is a point and so diam σ(∆p ) = 0 and since Bσ = σ by definition the
proposed inequality holds. For the inductive step, remember that Bσ = bσ (B∂σ) and by
p
(3.6) mesh Bσ = max{ p+1
diam σ(∆p ), mesh(B∂σ)} (since every simplex in ∂σ is contained
p
in σ) and by the induction hypothesis mesh(B∂σ) 6 p−1
p mesh ∂σ 6 p+1 mesh ∂σ.
Paragraph 4. Small Simplices and Standard Models
4.
23
Small Simplices and Standard Models
(4.1)
Definition. Let X be a topological space and U ⊂ PX a collection of subsets of
X such that the interiors of all U ∈ U cover X. A singular simplex σ : ∆p → X is called
P
U-small iff σ(∆p ) ⊂ U for some U ∈ U. More generally a p-chain c = m
i=1 ni σi is called
U-small iff all the σi are U-small. The subgroup of Sp X spanned by the U-small simplices
gives us an abelian group Sp U and this defines a subcomplex SU of SX (obviously the image
of a U-small p-chain under the boundary morphism is a U-small (p − 1)-chain).
As promised, with the last theorem of the last section we can finally apply Lebesgue’s
Lemma and conclude.
(4.2)
Corollary. Let X = ∆q be a standard model and U ⊂ PX a collection of subsets of
X, whose interiors form an open cover of X. For any singular simplex (σ : ∆p → X) ∈ Sp X,
there is a k ∈ N such that B k σ ∈ Sp U.
Proof. (σ −1 Ů )U ∈U is an open cover of ∆p and since this is compact there is a Lebesgue
number ε ∈ R>0 such that every V ⊂ X with diam V < ε is wholely contained in σ −1 U for
some U ∈ U. By (3.7) the proposition follows.
(4.3)
Remark. We have already seen in (2.2) that the functor S : Top → AbGrp is
free with models S and aacyclic on S. So, we can use corollary (2.4) of the last chapter to
conclude that the barycentric subdivision B : S → S is naturally homotopic to 1S . That is,
for each space X there is a chain homotopy DX : BX ' 1SX natural in X (we will again omit
the index if there is no risk of confusion). By the naturality of D it follows that DX maps SU
into itself for any cover U of X. To see this, let σ : ∆p → X be a U-small singular p-simplex,
σ
i
i.e. σ factors as ∆p −
→U →
− X for some U ∈ U, where i : U ,→ X is the inclusion. Since
i∗ ◦ DU = DX ◦ i∗ (this is the naturality of D) it follows that DX σ = DX ◦ i∗ σ = i∗ ◦ DU σ
and because i∗ : SU ,→ SX is again the inclusion DX σ ∈ SU.
(4.4)
Lemma. Let X be a topological space and U ⊂ PX a collection of subsets of X,
whose interiors form an open cover of X. Then the inclusion i : SU ,→ SX induces and
isomorphism i∗ : H(SU) ∼
= H(SX).
Proof. Clearly H0 (SU) = S0 U = S0 X = H0 (SX) and we have to prove that i∗ : Hp (SU) →
Hp (SX) is an isomorphism for all p > 1. For c ∈ Zp X = ker(∂ : Sp X → Sp−1 X) a p-cycle by
the last corollary (4.2) there is a k ∈ N such that B k c ∈ Sp U. Then B k c−c = Dk ∂c+∂Dk c =
∂Dk c and so [c] = [B k c] ∈ Hp X, which proves the surjectivity of i∗ . For injectivity, we must
prove that Bp (SX) = im(∂ : Sp+1 X → Sp X) ⊂ Bp (SU). I.e. we show that each p-chain
of SU, which is a boundary in SX is already a boundary in SU. So let c = ∂b0 ∈ Sp U
for some b0 ∈ Sp+1 X. By the last corollary (4.2) there is a k ∈ N such that B k b0 ∈ Sp+1 U
and by the above remark Dk c ∈ Sp+1 U and we put b := B k b0 − Dk c ∈ Sp+1 U and because
B k − 1SX = ∂Dk + Dk ∂, obviously
∂b = ∂B k b0 − ∂Dk c = B k ∂b0 − (B k c − c − Dk ∂c) = B k c − B k c + c + Dk ∂∂b0 = c.
With this lemma at hand we can finally embark on the task of proving the excision
axiom for singular homology.
Chapter 3. Singular Homology
24
5.
Excision
This section is solely devoted to proving the excision axiom for singular homology. Here our
investment into the machinery of homological algebra (in the form of the acyclic model theorem) pays off again and the proof boils down to simple algebra with no geometric arguments
whatsoever.
(5.1)
Definition. We define the category Cov having as objects all pairs (X, U) where
X is na topological
o space and U ⊂ PX is a collection of subsets of X, whose interiors
Ů := Ů U ∈ U cover X. An arrow f : (X, U) → (Y, V) consists of a map f : X → Y such
that U = f −1 V := f −1 V V ∈ V . We equip this category with the models
n
o
M := (∆q , U) q ∈ N, U ⊂ PX arbitrary, such that Ů covers X .
We define two functors S 0 , S 00 : Cov → Ch(AbGrp) on the objects by S 0 (X, U) := SX,
S 00 (X, U) := SU and and the arrows by S 0 f = S 00 f = f∗ .
We are now able to prove our main theorem. The proof is essentially the same as the
proof that S : Top → Ch(AbGrp) is free and aacyclic on the standard models (cf. remark
(2.2)) but is still given in full detail.
(5.2)
Theorem. S 0 and S 00 are both free with models M and aacyclic on M.
Proof. We first check the freeness, which we have to check at each degree p ∈ N. We first
choose
n
o
Mp := (∆p , U) U ⊂ PX, such that Ů covers ∆p ⊂ M
and ep := 1S∆p ∈ S 0 (∆p , U) = S∆p for (∆p , U) ∈ Mp . For an arbitrary object (X, U) in
Cov and σ : ∆p → X and arbitrary singular p-simplex in Sp X there is one and only one
cover, namely V := σ −1 U, of ∆p such that σ defines an arrow σ : (∆p , V) → (X, U). So
σ ∈ Cov ((∆p , V), (X, U)) and
{σ∗ ep = σ | (∆p , V) ∈ Mp , σ ∈ Cov ((∆p , V), (X, U))}
is a basis for Sp X and so S 0 is free with models M. By the same argument (with ep := 1SU
instead of 1S∆p ) S 00 is free with models M.
Clearly S 0 is aacyclic on M and so is S 00 by the lemma (4.4) above, which proves
our proposition.
Again by the above lemma (4.4) there is a natural isomorphism ϕ : H0 S 00 ∼
= H0 S 0
defined by ϕ(X,U ) := i∗ : H0 (SU) → H0 (SX), where i : SU ,→ SX is the inclusion. So by
∼ S 0 and as an
corollary (2.3) in chapter 2 this can be lifted to a natural isomorphism ϕ̃ : S 00 =
immediate consequence we get the following corollary.
(5.3)
Corollary. For X a topological space and U ⊂ PX such that Ů covers X, the
inclusion SU ,→ SX is a chain equivalence.
(5.4)
Corollary. (Excision for Singular Homology) Let (X, A) ∈ Ob(Top(2) ) be a
pair, U ⊂ X open with U ⊂ Å. Then the standard inclusion i : (X \ U, A \ U ) ,→ (X, A)
induces an isomorphism Hn (X \ U, A \ U ) ∼
= Hn (X, A) for all n ∈ N.
Paragraph 5. Excision
25
Proof. Consider U := {A, X \ U } which satisfies the condition that Ů covers X. By definition
SU = SA + S(X \ U ) and the inclusion SU = SA + S(X \ U ) ,→ SX is a chain equivalence by
the last corollary. So the inclusion SU/SA ,→ SX/SA = S(X, A) is also a chain equivalence.
Now notice that S(A \ U ) = S(X \ U ) ∩ SA and so the inclusion S(X \ U ) ,→ SX induces an
isomorphism
∼ SU/SA = (SA + S(X \ U ))/SA.
S(X \ U )/S(A \ U ) =
All in all, the inclusion from the proposition induces a chain equivalence S(X \ U, A \ U ) '
S(X, A), which gives us an isomorphism in homology.
LITERATURE
Dold, A.: Lectures on Algebraic Topology, Springer, Berlin; Heidelberg; New York, 1980
Eilenberg, S.; Steenrod, N.: Foundations of Algebraic Topology, Princeton University
Press, Princeton, 1952
Heistad, E.: Excision in Singular Theory, Mathematica Scandinavica 20, 61–64, 1967
Hilton, P.: A Brief, Subjective History of Homology and Homotopy Theory in This Century, Mathematics Magazine 61, 282–291, 1988
Spanier, P.: Algebraic Topology, Springer, Berlin; Heidelberg; New York, 1994
Stöcker, R.; Zieschang, H.: Algebraische Topologie, Teubner, Stuttgart, 1988
tom Dieck, T.: Topologie, De Gruyter Lehrbuch, Berlin, 2000
Vick, J. W.: Homology Theory: An Introduction to Algebraic Topology, Springer, New
York, 1994
INDEX
Numbers
Cone, 19
Connecting morphism, 5
1, 4
A
Aacyclic chain complex, 14
Aacyclic functor
on models, 14
Absolute homology, 5
Absolute singular homology, 18
Acyclic Model Theorem, 14
Admissible
category for homology theory, 4
Affine singular simplex, 19
Almost acyclic functor
on models, 14
B
Barycentre, 20
Barycentric subdivision, 20
Boundary, 17
Boundary morphism, 17
Boundary operator, 5
Braid diagram, 8
Braid Lemma, 8
C
Category
admissible for homology theory, 4
Chain, 17
affine, 21
U-small, 23
Chain complex
aacyclic, 14
Chain homotopic
natural transformations, 14
Coefficients
of a homology theory, 6
D
Dimension
invariance of, 12
E
Excision, 6
for singular homology, 24
F
Face
of a singular simplex, 17
Free functor
with models, 13
H
Homology
of a pair, 5
of a space, 5
of spheres, 12
reduced, 9
Homology sequence, 6
of a triple, 9
reduced, 10
Homology theory, 5
Homotopic
maps in Top(2) , 5
natural transformations, 14
Homotopy
in Top(2) , 5
Homotopy Invariance, 5
Homotopy invariance
of singular homology, 18
28
I
I, 4
Inclusion, 3
Invariance
of dimension, 12
L
Lattice
of a pair, 4
Long Exact Homology Sequence, 6
of a triple, 9
reduced, 10
Index
relative, 18
Singular p-chain, 17
Singular simplex, 17
Space, 3
Standard models, 19
Standard simplex, 17
Subdivision
barycentric, 20
T
Top, 3
Top(2) , 3
Top(3) , 3
M
U
Map, 3
Models
of a category, 13
U-small chain, 23
U-small simplex, 23
V
N
v-cone, 19
Naturally chain homotopic, 14
Naturally homotopic, 14
O
Ordinary
homology theory, 6
R
Reduced homology, 9
Relative homology, 5
of discs, 12
Relative singular homology, 18
Restriction functor, 5
S
Shift functor, 14
Simplex, 17
affine singular, 19
standard, 17
U-small, 23
Singular homology, 18
absolute, 18