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Page 1: Homology Theory - EPFLsma.epfl.ch/~werndli/scripts/homology/homology.pdf · homology theory and all spaces and maps are assumed to be admissible (i.e. lie in C). As ... By the long

Homology TheoryKay Werndli

13. Dezember 2009

This work, as well as all figures it contains, is licensed under a Creative Commons CC© BY:© $\© C©Attribution-Noncommercial-Share Alike 2.5 Switzerland License. A copy of the license text may be foundunder http://creativecommons.org/licenses/by-nc-sa/2.5/ch/deed.en_GB.

Page 2: Homology Theory - EPFLsma.epfl.ch/~werndli/scripts/homology/homology.pdf · homology theory and all spaces and maps are assumed to be admissible (i.e. lie in C). As ... By the long

CONTENTS

Chapter 1 Axiomatic Homology Theory . . . . . . . . . . . . . . . . 31. The Eilenberg-Steenrod Axioms . . . . . . . . . . . . . . . . . . . . 32. First Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 63. Reduced Homology . . . . . . . . . . . . . . . . . . . . . . . . . 94. Homology of Spheres . . . . . . . . . . . . . . . . . . . . . . . . 10

Chapter 2 Acyclic Models . . . . . . . . . . . . . . . . . . . . . . 131. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132. The Acyclic Model Theorem . . . . . . . . . . . . . . . . . . . . . 14

Chapter 3 Singular Homology . . . . . . . . . . . . . . . . . . . . 171. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172. Homotopy Invariance . . . . . . . . . . . . . . . . . . . . . . . . 193. Barycentric Subdivision . . . . . . . . . . . . . . . . . . . . . . . 194. Small Simplices and Standard Models . . . . . . . . . . . . . . . . . 235. Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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Chapter 1

AXIOMATIC HOMOLOGY THEORY

[Der Satz] verhüllt die geometrische Wahrheit mit dem Schleierder Algebra.

TAMMO TOM DIECK

Homology theory has been around for about 115 years. It’s founding father was thefrench mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homol-ogy” should be in 1895. Thirty years later, it was realised by Emmy Noether that abeliangroups were the right context to study homology and not the then known and extensivelyused Betti numbers. In the decades after the advent of Poincaré’s homology invariants, manydifferent theories were developed (e.g. simplicial homology, singular homology, Čech homol-ogy 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 whenSamuel 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) thecategories of topological spaces, pairs of spaces (called “pairs” for short), and triples of spacesrespectively. I.e. the objects of Top(2) are pairs (X,A), where X ∈ Ob(Top) is a topologicalspace and A ⊂ X and a morphism f : (X,A)→ (Y,B) is a continuous map f : X → Y withfA ⊂ B. Analogously the objects of Top(3) are triples (X,A,B), where X ∈ Ob(Top) andB ⊂ A ⊂ X and a morphism f : (X,A,B)→ (Y,A′, B′) is a continuous map f : X → Y withfA ⊂ A′ and fB ⊂ B′. 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 identificationthroughout 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 notsurprising that these two types of inclusions constitute to two adjunctions. If we denote the

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4 Chapter 1. Axiomatic Homology Theory

first inclusion by F and the second one by G the adjunctions are as follows:

Top(2)U // TopG

oo _ and TopF // Top(2)U

oo _ ,

where U : Top(2) → Top is the forgetful functor (X,A) 7→ X (similarly for Top(3) andTop(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(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 throughoutthe whole text.

(1.2) Definition. A subcategory C 6 Top(2) is called admissible for homology theoryiff

(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) and1 ∼= {∗}, we have

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,∅)

$$JJJJJJJJJ

(∅,∅) // (A,∅)

::uuuuuuuuu

$$IIIIIIIII(X,A) // (X,X) .

(A,A)

::ttttttttt

Moreover, we require that for f : (X,A) → (Y,B) in C, C also contains all themaps 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

(X,A)ι0 //

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 theinclusion (X,∅) → (X,A). Moreover, it follows that C contains I since C contains 1 and1× I ∼= I.

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Paragraph 1. The Eilenberg-Steenrod Axioms 5

(1.4) Example. The following categories are all examples of admissible categories forhomology 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 locallycompact Hausdorff and A ⊂ X is closed and as arrows all maps of pairs, satisfyingthat 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 f0x = f(x, 0) and f1x = f(x, 1). That means that f is an ordinary homotopyfrom 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 looselyrefer to this family of maps as a homotopy from f0 to f1. As always, we call f0 and f1 asabove 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 adiagram of the form

(X,A)ι0 //

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 functorρ : C → C which sends (X,A) to (A,∅) and f : (X,A) → (Y,B) to ρf =: f |AB : (A,∅) →(B,∅), x 7→ fx. This functor is well-defined by axiom (ii) in the definition of an admissiblecategory.

(1.7) Definition. A homology theory on an admissible category C consists of a family offunctors (Hn : C→ A)n∈Z, where A is an abelian category and a family of natural trans-formations (∂n : Hn → Hn−1 ◦ ρ)n∈Z. Hn(X,A) is called the nth homology of (X,A) and ∂nthe nth boundary operator or connecting morphism. As mentioned before, we identify X with(X,∅) and in the same spirit write HnX or Hn(X) for Hn(X,∅), which we call the nth

(absolute) homology of X. To avoid unnecessarily complicated notation, we write f∗ forHnf : Hn(X,A) → Hn(Y,B) where f : (X,A) → (Y,B) and we will usually omit the indexand write ∂ for ∂n. Explicitly, ∂ being a natural transformation means that the followingdiagram commutes for all f : (X,A)→ (Y,B) in C.

(X,A)

f

��

Hn(X,A)

f∗��

∂ // Hn−1A

f∗��

(Y,B) Hn(Y,B)∂

// Hn−1B .

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)∗.

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6 Chapter 1. Axiomatic Homology Theory

(ii) (Long Exact Homology Sequence) For each (X,A) ∈ Ob(C) we have a longexact sequence

. . .→ Hn+1(X,A) ∂−→ HnA→ HnX → 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 ⊂ A and the standardinclusion (X \ U,A \ U) → (X,A) lies in C. Then this inclusion induces for eachn ∈ 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−10 ⊂ f−1 [0, 1[ ⊂ A and the inclusion (X \U,A \U)→ (X,A) liesin 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 Hn1 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

Hn1 = Hn(1,∅) = 0 ∀n ∈ Z \ {0}.

So in an ordinary homology theory only the coefficientH01 is of any interest. If we have chosenan isomorphism H01 ∼= G ∈ A we call this an ordinary homology theory with coefficients inG 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). Asa 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 homologysequence 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.

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Paragraph 2. First Consequences 7

(b) If f : A→ X is a homotopy equivalence then f∗ : HnA→ HnX is an isomorphism.In particular if A is a deformation retract of X (i.e. the inclusion i : A ↪→ X is ahomotopy equivalence) then i∗ : HnA → HnX is an isomorphism and Hn(X,A) =0.

More generally, one immediately deduces the following from part (b) of the lastremark, 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 |AB : A → B arehomotopy 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 thedual situation (i.e. the homology of a product) which would lead to the definition of theso-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 qX2, A1 qA2) 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

Hn(X1, A1)(i1)∗−−−→ Hn(X,A) (i2)∗←−−− 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.

Proof. Consider the morphism(

(i1)∗(i2)∗

)from the direct sum of the long exact homology se-

quences for (X1, A1) and (X2, A2) to the long exact homology sequence for (X,A). In viewof the 5-lemma it is enough to show the proposition for the case where A1 = A2 = ∅. Wehave the standard inclusions

X1i1−→ X

j1−→ (X,X1) and X2i2−→ X

j2−→ (X,X2),

whose induced morphisms can be combined in a commutative diagram

HnX1

(i1)∗ $$HHHHHHHHHf1

// Hn(X,X2)

HnX

(j2)∗

88rrrrrrrrrr

(j1)∗

&&LLLLLLLLLL

HnX2

(i2)∗::vvvvvvvvv

f2// Hn(X,X1) .

By the long exact homology sequences the diagonals are exact and by the excision axiomany morphism of the form Hn(Y,B) → Hn(Y q Z,B q Z), induced by the inclusion, is anisomorphism. So in particular, f1 and f2 are isomorphisms. The following lemma gives thedesired result. �

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8 Chapter 1. Axiomatic Homology Theory

(2.4) Lemma. Given a commutative diagram

A1

i1 $$IIIIIIf1

// B2

Xj2

::uuuuuu

j1

$$IIIIII

A2

i2::uuuuuu

f2// B1

in an abelian category with exact diagonals. Then the following conditions are equivalent:

(a) f1 and f2 are isomorphisms;

(b)(i1i2

): 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 whetherthere is an analogue for a triple (X,A,B) and indeed there is. There are topological proofsfor this but we prefer an algebraic one (even if that means that we have to draw a nastydiagram) 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 defineanother boundary operator

∂ : Hn+1(X,A) ∂−→ HnA→ Hn(A,B),

where the first morphism is the boundary map given by our homology theory and the sec-ond morphism is induced by the inclusion (A,∅) → (A,B). We shall also write ∂ for thismorphism 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 longexact homology sequences into a so-called braid diagram

(1)

&&MMMM

Hn+1(X,A)∂

&&MMMMMM

qqqqqqqqq &&MMMMMMMMM

Hn(A,B)&&MMMMM

qqqqqqqqq &&MMMMMMMMM

Hn−1B

&&MMMMMM

qqqqqqqqq &&MMMMMMMM

(2)88qqqq

(3) ∂

&&MMMMHnA

&&MMMMMM

88qqqqqqHn(X,B)

&&MMMMM

∂88qqqqq

Hn−1A

88qqqq

&&MMMM

HnB

88qqqqqq

MMMMMMMMM 88qqqqqqqqq

HnX

MMMMMMMMM 88qqqqqqqqq

88qqqqqqHn(X,A)

∂88qqqqq

∂MMMMMMMMM 88qqqqqqqq(4)

∂ 88qqqq,

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 sequencefor the triple (X,A,B). One easily checks that this is a chain complex (i.e. the compositionof 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 abeliancategory (where the homologies are replaced by arbitrary objects of this category), wherethree of the sequences are exact and the fourth is a chain complex then this will be exact,too.

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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 calledthe long exact homology sequence of the triple (X,A,B).

3. Reduced HomologyAlthough the coefficients Hn1 are important, they do not contain any geometric informationwhatsoever. Because of this and for the sake of readability (so that we do not always have tocarry these one-point spaces with us while doing algebraic manipulations), we want to splitthem 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 aone-point space. We define the nth reduced homology of X as

HnX := ker (p∗ : HnX → Hn1) .

For f : X → Y , we get again an induced morphism f∗ : HnX → HnY in the obvious way.Like that we can extend Hn to a homotopy invariant functor Top→ A (i.e. if f is a homotopyequivalence, 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 otherway. And indeed by some elementary algebraic facts we can. If we choose a point x ∈ Xand look at the inclusion x : 1 → X and p : X → 1, we have p ◦ x = 11 and the long exacthomology sequence for (X,x) reads as

. . .→ Hn+1(X,x)∂−→ Hn1

x∗−→ HnX → Hn(X,x)∂−→ Hn−11

x∗−→ . . . .

Because p∗ ◦x∗ = 1Hn1 it follows that x∗ is a monomorphism and by exactnesss im ∂ = 0. Sowe can rewrite this as a short exact sequence, which splits since p∗ is a retraction of x∗.

0 // Hn1

1Hn1 ##GGGGGGGGGx∗ // HnX

p∗

��

// Hn(X,x) // 0

Hn1 .

The triangle on the left gives us an isomorphism(ix∗

): HnX ⊕ Hn1

∼−→ HnX, where i :HnX ↪→ HnX is the standard inclusion. One plainly checks this as follows:

HnX ∼= ker p∗ ⊕ x∗(Hn1) = HnX ⊕ x∗(Hn1) ∼= HnX ⊕Hn1.

By the first isomorphism theorem j∗|HnX : HnX∼−→ Hn(X,x) is an isomorphism, where

j : X → (X,x) is the standard inclusion:

Hn(X,x) ∼= HnX/x∗(Hn1) ∼=(HnX ⊕ x∗(Hn1)

)/x∗(Hn1) ∼= HnX

and finally these two together give

HnX ∼= Hn(X,x)⊕Hn1 by HnX

(ix∗

)←−−− HnX⊕Hn1

j∗|HnX×1Hn1−−−−−−−−−→ Hn(X,x)⊕Hn1,

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10 Chapter 1. Axiomatic Homology Theory

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 exactfrom which one easily deduces the following theorem.

(3.2) Theorem. If X is contractible, then HnX ∼= Hn(X,x) = 0 ∀n ∈ Z. �

To finish this section, we are going to introduce the analogue of the long exacthomology sequence in the reduced case. As one easily verifies, this is just a special case ofthe long exact homology sequence for a triple, where the triple is of the form (X,A, x), wherex ∈ A ⊂ X is a point.

(3.3) Theorem. (Reduced Long Exact Homology Sequence) Let (X,A) be a pairwith A 6= ∅. Then the image of the boundary operator ∂ : Hn+1(X,A)→ HnA lies in HnA.As a consequence, by restricting the long exact homology sequence of the pair (X,A), we getanother long exact sequence for the reduced homology

. . .→ Hn+1(X,A) ∂−→ HnA→ HnX → Hn(X,A) ∂−→ . . . .

Proof. Consider the unique arrow p : X → 1 (resp. p : A → 1 or p : (X,A) → (1, 1)). Thenthe long exact sequences of (X,A) and (1, 1) yield

. . . // Hn+1(X,A) ∂ //

��

��

HnA//

��

��

HnX//

��

��

Hn(X,A) ∂ //

��

��

. . .

. . . // Hn+1(X,A) ∂ //

p∗����

HnA //

p∗����

HnX //

p∗����

Hn(X,A) ∂ //

p∗����

. . .

. . .0

// Hn+1(1, 1) 0// Hn1 ∼

// Hn1 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 thatHn1 → Hn1 is an isomorphism. The upper row consists simply of the kernels of the cor-responding 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 (thisactually 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 chooseany point x : 1 → A (resp. x : 1 → X or x : (1, 1) → (X,A)), we have that p ◦ x = 11 andso p has a section. By a general theorem (whose proof is left as an exercise) which says thatif we have an epimorphism of exact sequences then its kernel is also exact (and dually for amonomorphism of exact sequences and its cokernel) we get the exactness of the reduced longexact homology sequence. �

4. Homology of SpheresIn this section, we delve into the problem of calculating the homology of the spheres justfrom the axioms. To do so, we observe first, that we can divide the sphere Sn ⊂ Rn+1 in anupper and lower hemisphere

Dn± := {(x1, . . . , xn+1) ∈ Sn | ±xn+1 > 0} .

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Paragraph 4. Homology of Spheres 11

Obviously Dn±∼= Dn by simply projecting Dn

± along the xn+1-axis to Rn × {0} ⊂ Rn+1.Moreover, we observe that we have for any n ∈ N an inclusion

Sn−1 → Sn

(x1, . . . , xn) 7→ (x1, . . . , xn, 0)

or a little more geometric, by viewing Sn−1 as the equator of Sn. By combining these we getS0 ↪→ S1 ↪→ S2 ↪→ . . . . Let’s furthermore fix the notation ei ∈ Rn to denote the ith standardbasis vector having (ei)j = δi,j (the Kronecker delta) and with this, let’s write N := en+1and S := −en+1 for the north and south pole of Sn respectively. Now, for n ∈ N>0 we lookat the commutative diagram

Hk(Dn−, S

n−1) //

��

Hk(Sn, Dn+)

��

Hk(Dn−, D

n− \ {S}) // Hk(Sn, Sn \ {S})

induced by inclusions. Because Sn−1 ↪→ Dn \ {S} and Dn+ ↪→ Sn \ {S} are homotopy

equivalences, it follows that the vertical arrows are isomorphisms (by remark (2.2)). For thebottom arrow, we can use the excision axiom with U := Dn

+ and deduce that this is also anisomorphism. In conclusion, the top arrow has to be an isomorphism, too.

We choose ∗ := e1 = (1, 0, . . . , 0) ∈ Sn−1 ⊂ Sn and insert this isomorphism in asecond diagram

Hk(Dn−, S

n−1)

o��

∂ // Hk−1(Sn−1, ∗)σ+

��

∼= Hk−1Sn−1

σ+

��

Hk(Sn, Dn+) Hk(Sn, ∗)j

oo ∼= HkSn

The reduced long exact homology sequence of (Dn−, S

n−1) reads as

. . .→ Hk(Dn−)→ Hk(Dn

−, Sn−1) ∂−→ Hk−1(Sn−1)→ Hk−1(Dn

−)→ . . . .

Since Dn−∼= Dn−1 is contractible, we deduce from theorem (3.2) that Hk(Dn

−) = Hk−1(Dn−) =

0 and so ∂ is an isomorphism. By the same argument, j is an isomorphism, since {∗} ↪→ Dn− is

a homotopy equivalence. Now we can easily define σ+ to be the unique isomorphism makingthe diagram commute.

(4.1) Lemma. HnS0 ∼= Hn1 for all n ∈ Z.

Proof. Choose a point x : 1 → S0 and denote the other point by y : 1 → S0. Let’s alsowrite i : HnS

0 ↪→ HnS0 for the standard inclusion. As seen in the section about the reduced

homology and theorem (2.3), we get a commutative diagram

HnS0 ⊕Hn1

(ix∗

)∼

// HnS0 Hn1⊕Hn1

( y∗x∗ )∼

oo

Hn1OO

OO

Hn1OO

OO

,

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12 Chapter 1. Axiomatic Homology Theory

where the vertical arrows are the standard inclusions into the second summand. If we definethe isomorphism f := ( y∗x∗ )

−1 ◦(ix∗

): HnS

0 ⊕Hn1∼−→ Hn1 ⊕Hn1, we can rewrite this as a

commutative diagram

0 // Hn1 // //

o

HnS0 ⊕Hn1 // //

f o��

HnS0 //

f��

// 0

0 // Hn1 // // Hn1⊕Hn1 // // Hn1 // 0 ,

where the rows are exact and f 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 forshort exact sequences) it follows that f is an isomorphism. �

(4.2) Theorem. For all k ∈ Z and n ∈ N we have isomorphisms

HkSn ∼= Hk−n1 and HkS

n ∼= Hk−n1⊕Hk1.

It follows that we also have isomorphisms

Hk(Dn+1, Sn) ∼= Hk−1Sn ∼= Hk−(n+1)1.

Proof. As seen in the last paragraph, we have isomorphisms

HkSn ∼= Hk−1S

n−1 ∼= . . . ∼= Hk−nS0 ∼= Hk−n1,

where we used the above lemma for the last isomorphism. By remembering ourselves thatHkS

n ∼= HkSn⊕Hk1 the first part of the proposition follows. For the second part, let’s look

at the reduced long exact homology sequence for (Dn+1, Sn). Because Dn+1 is contractible,this looks like

0→ Hk(Dn+1, Sn) ∂−→ Hk−1Sn → 0

and so ∂ is an isomorphism. �

(4.3) Corollary. Let (Hn)n∈Z, (∂n)n∈Z be an ordinary homology theory, having coeffi-cient H01 ∼= G then for n ∈ N>0

HkSn ∼=

{G k ∈ {0, n}0 otherwise and similarly Hk(Dn+1, Sn) ∼=

{G k = n+ 10 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 wewe have a homeomorphism Rm → Rn. The case where m = 0 or n = 0 is trivial an so thecase m,n > 1 is left. We can extend our homeomorphism Rm → Rn to a homeomorphism ofthe one-point compactifications, which is Sm ∼= Sn but since m 6= n by the above theoremHnS

m = 0 but HnSn ∼= G 6= 0. �

(4.5) Corollary. Sn is not contractible ∀n ∈ N �

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Chapter 2

ACYCLIC MODELS

In der Algebra gibt es viele Definitionen; manche werden auchgebraucht.

ARMIN LEUTBECHER

In this chapter we will be concerned with studying so-called acyclic models and willprove a form of the famous acyclic model theorem. The theory of acyclic models is in somesense a way to abstract the standard models arising in homology theory, like the standardsimplices in singular homology (see the next chapter). The acyclic model theorem will beuseful in this text to prove homotopy invariance and the excision axiom for singular homologybut generally finds wide applications throughout algebraic topology and homological algebra.

1. Models

Let C be a category. A specified setM⊂ Ob(C) of objects in C will be called models of C.From now on we fix the notationM 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. Andin 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 withmodels M iff there is a subset M′ ⊂ M and for each M ∈ M′ an element eM ∈ FM suchthat for every C ∈ Ob(C) the module FC is free and the set{

(Fa)eM ∈ FC∣∣M ∈M′, a ∈ C(M,C)

}forms a basis for FC. Put differently, the functor F factors as

Sets

��??????

C

??������

F// R-Mod ,

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∗ ((Fa)eM ) := F (ba)eM . If F doesmap to Ch(R-Mod) (the category of chain maps between chain complexes of R-modules) wecall F free with models M iff it is free with models M at each degree Fp, where for p ∈ ZFp : C → R-Mod maps a : C → D to (Fa)p : (FC)p → (FD)p. One should notice that at

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14 Chapter 2. Acyclic Models

each degree, we can have a different subsetMp ⊂ M and different elements epM ∈ FnM forM ∈Mp.

Finally, let’s call a functor F : C→ Ch(R-Mod) aacyclic on the modelsM (aacyclicstands for almost acyclic) iff for each M ∈ M the chain complex FM is exact everywhereexcept at the degree 0. I.e. Hi(FM) = 0 ∀i ∈ Z \ {0}. In the same spirit, we call a chaincomplex 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 twonatural transformations. We say that α and β are naturally chain homotopic or simplynautrally homotopic iff all their components are chain homotopic in a natural way. That isfor each C ∈ Ob(C) there is a chain map χC : FC → GC of degree 1 (i.e. (χC)n : FnC →Gn+1C ∀n ∈ Z) such that

αC − βC = ∂GC ◦ χC + χC ◦ ∂FC ,

where ∂FC and ∂GC denote the boundaries of FC and GC respectively. Furthermore, χC isrequired to be natural in C. That is for each a : C → D in C the following diagram commutes

C

a

��

FC

Fa��

χC // GC

Ga��

D FD χD// GD .

So in some sense χ is a natural transformation F → G, which is not completely honest sincethe components χC are not really arrows in Ch(R-Mod). To be even more formal, one couldsay 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 ′ = S−X,having X ′n = Xn+1 with the obvious boundaries (the arrow function of S− is obvious, too).

2. The Acyclic Model TheoremIn 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 modelsM andF,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 modelsM andG is aacyclic onM and there is a natural transformationϕ : H0F → H0G (where H0F,H0G : C → R-Mod) then there is a natural transformationϕ : F → G, which induces ϕ. Moreover, ϕ is unique up to natural homotopy.

(2.2) Remark. For the sake of readability we will omit unnecessary indices in the follow-ing proof. We will assume that the attentive reader will still be capable of following it andfill in the details.

This proof seems a little complicated at first glance (which it really isn’t). Becauseof this we will briefly sketch it before formalizing it rigorously. For C ∈ Ob(C), we want todefine ϕ as to make the diagram

. . . ∂ // Fn+1C∂ //

ϕ

��

FnC∂ //

ϕ

��

. . . ∂ // F0C // //

ϕ

��

H0(FC) //

ϕ

��

0

. . .∂

// Gn+1C∂

// GnC ∂// . . .

∂// G0C // // H0(GC) // 0

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Paragraph 2. The Acyclic Model Theorem 15

commute. We do this inductively in each degree n. To do so, we first consider the case whereC = M is a model and so the lower row is exact. We can use this exactness to “lift” ϕ fromdegree n to n + 1. Afterwards we use the fact that F is free with modelsM to extend thisto arbitrary C ∈ Ob(C).

Proof. By presumption, for each n ∈ N there is a collectionMn ⊂M and for each M ∈Mn

an element enM ∈ FnM as in the definition of a free functor with models M. Now, letC ∈ Ob(C) be arbitrary. Again by presumption, F0C = Z0(FC) and G0C = Z0(GC) arethe cycles of FC and GC at degree 0 respectively. Thus, we get standard projections ontothe 0th homology modules as in the following diagram

F0C

ϕ

��

// // H0(FC)

ϕ

��

G0C // // H0(GC) .

Thus, we can augment F and G by (re)defining F−1C := H0(FC) and G−1C := H0(GC)and defining the standard projections as the boundary morphisms. By this, ϕ is defined indegree −1, where it is simply ϕ.

For the inductive step let’s assume that ϕ is defined in degree n− 1 for n > 0. Foreach model M ∈Mn we consider ϕ(∂enM ) ∈ Gn−1M (which we have already defined). Sincethe diagram

FnM

ϕ

��

∂ // Fn−1M∂ //

ϕ

��

Fn−2M

ϕ

��

GnM ∂// Gn−1M

∂// Gn−2M .

commutes and the lower row is exact (because G is aacyclic on M) we conclude that∂ϕ(∂enM ) = ϕ(∂∂enM ) = 0 and so ϕ(∂enM ) ∈ Gn−1M must be a boundary. I.e. we canchoose c ∈ GnM satisfying ∂c = ϕ(∂enM ) and define ϕenM := c, which makes the abovediagram commute.

For a : M → C a morphism in C (Fna)enM ∈ FnC is a basis element of FnC and wedefine ϕ ((Fna)enM ) := (Gna)ϕenM . We do this for every a and every M and like that define ϕon the basis elements of FnC which means that we can extend it uniquely to ϕ : FnC → GnC.To check that ϕ thus defined is a chain morphism (i.e. commutes with the boundaries), welook at the following cubical diagram

FnMϕ

//

��

Fna

yyrrrrrrrrGnM

��

Gna

yyrrrrrrrr

FnCϕ

//

��

GnC

��

Fn−1Mϕ

//

Fn−1ayyrrrrrrrrGn−1M

Gn−1ayyrrrrrrrr

Fn−1Cϕ

// Gn−1C ,

where the downward arrows are all boundary morphisms. The left and right faces of thecube obviously commute and the bottom commutes by inductive hypothesis. For the elementenM ∈ FnM the top and the back faces commute by definition of ϕ. So the front face has

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16 Chapter 2. Acyclic Models

to commute, too for (Fna)enM ∈ FnC. Since this holds for all M and all a the front facecommutes for all basis elements of FnC and so commutes as a whole. This defines ϕ indegree n. One easily checks the naturality of ϕ, i.e. for a : C → D an arrow in C theequation ϕ ◦ Fa = Ga ◦ ϕ holds.

What is left to prove is the uniqueness up to natural homotopy. So suppose thatϕ, ψ : F → G are two natural transformations inducing ϕ in the 0th homology or put differ-ently, that lift ϕ (defined in degree −1). For each object C ∈ Ob(C) we must define a chainhomotopy χ : FC → GC from ϕ to ψ (i.e. χ is of degree 1 and satisfies ∂χ + χ∂ = ϕ − ψ),which is natural in C. χ is already defined in degree −1 (note that we are still working withthe augmented F and G), where it is simply 0 because there ϕ = ψ = ϕ. Suppose now, thatχ is defined in degree n− 1 with n > 0. FnC has {(Fa)enM}M,a as a basis and we notice thatb := ϕenM − ψenM − χ(∂enM ) is a cycle because

∂ϕenM − ∂ψenM − ∂χ∂(enM ) = ϕ(∂enM )− ψ(∂enM )−(−χ(∂∂enM ) + ϕ(∂enM )− ψ(∂enM )

)= 0.

Because GM is aacyclic, b must be a boundary, i.e. there is a c ∈ Gn+1M with ∂c = b andwe define χenM := c. By defining χ ((Fna)enM ) := (Gn+1a)χenM we have defined χ on all basiselements and thence can extend it uniquely to χ : FnC → Gn+1C. One can easily check thatχ thus defined is really a chain morphism of degree 1 by using a cubical diagram, similar tothe one above. By construction this gives us a chain homotopy χ : ϕ ' ψ and it is plain tocheck naturality in C. �

(2.3) Corollary. If F,G : C→ Ch(R-Mod) are functors which are 0 in negative degreesand both free and aacyclic onM and there is a natural isomorphism ϕ : H0F ∼= H0G, thenϕ can be extended to a natural isomorphism ϕ : 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 withmodels M and aacyclic on M and α : F → F is a natural endotransformation inducingthe identity in 0th homology. Then α is naturally homotopic to 1F . In particular, for eachC ∈ Ob(C) there is a chain homotopy αC ' 1FC (i.e. αC and 1FC are chain homotopic).

Proof. By the acyclic model theorem there is a natural transformation ϕ : F → F whichinduces 1H0F : H0F → H0F and is unique up to homotopy. But α and 1F are two suchnatural transformations and so α and 1F are naturally homotopic. �

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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 toprove the axioms for an ordinary homology theory in the case of singular homology.

1. DefinitionsWe repeat the definition of singular homology in this section. We do so, mainly, to fix thenotation but we will also prove as a first result that the singular homology of contractiblespaces vanishes in positive degrees.

(1.1) Definition. For p ∈ N let ei ∈ Rp+1 be the i-th standard basis vector. We definethe p-dimensional standard simplex as

∆p :={(t0, . . . , tp) = t0e0 + . . .+ tpep ∈ Rp+1

∣∣∣ t0 + . . .+ tp = 1 and ti > 0 ∀i}.

Every function α : {0, . . . , p} → {0, . . . , q} induces an affine map

∆α : ∆p → ∆q,p∑i=0

tiei 7→p∑i=0

tieαi.

Especially interesting for our theory is the injective map δpi : {0, . . . , p− 1} → {0, . . . , p} fori ∈ {0, . . . , p} which simply leaves out the value i. We define dpi := ∆δpi and will usuallyomit the upper index where it is unnecessary. For X a topological space, we call a mapσ : ∆p → X a singular p-simplex in X and call σ ◦ dpi its ith face. Furthermore, for eachp ∈ Z we define a functor Sp : Top → AbGrp, which is 0 for p < 0 and otherwise sendsa space X to the free abelian group over all singular simplices σ : ∆p → X and a mapf : X → Y to f∗ : SpX → SpY which is defined on the basis elements of SpX by f∗σ := f ◦σ.An element of SpX is called a singular p-chain. The faces of a singular simplex σ : ∆p → Xcan be used to define a morphism

∂p : SpX → Sp−1X,σ 7→p∑i=0

(−1)iσ ◦ dpi

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

δp+1j ◦ δpi = δp+1

i ◦ δpj−1 ∀p ∈ N, i ∈ {0, . . . , p}, j ∈ {0, . . . , p+ 1}, i < j

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18 Chapter 3. Singular Homology

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+1X

∂−→ SpX∂−→ . . . and a map f : X → Y

to f∗ : SpX → SpY in each degree. We write HpX := Hp(SX) and call this the pth

singular homology of X (with coefficients in Z). As is well known from a basic course abouthomological algebra f∗ : SX → SY induces a morphism f∗ : HX → HY and this gives usa functor H : Top → Ch(AbGrp) (recall that the Hp(SX) form a chain complex with allboundary morphisms being 0). This gives us the absolute homology of a space X but to bea full-fledged homology theory, we must define relative homology, too. So if i : A ↪→ X is aninclusion, we define S(X,A) := SX/i(SA) = SX/SA = coker(i∗ : SA ↪→ SX), which definesa functor S : Top(2) → Ch(AbGrp) again by sending f : (X,A)→ (Y,B) to the morphismf∗ : SX/SA→ SY/SB induced by f∗ : SX → SY , the former being well-defined as fA ⊂ B.As usual, we call Hp(X,A) := Hp(S(X,A)) the pth singular homology of X relative to Aand 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 chaincomplexes SA // // SX // // S(X,A) induced by the standard inclusions and again by elemen-tary 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)) thestandard inclusions A ↪→ X ↪→ (X,A) yield a long exact sequence

. . .∂−→ HpA→ HpX → Hp(X,A) ∂−→ Hp−1A→ . . .→ H0(X,A)→ 0. �

To prove the homotopy invariance and excision axiom for the singular theory, it willbe 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. HpX = 0 ∀p 6= 0).Furthermore, H0X 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 constantmap. 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 elementof SpX) and define Dσ : ∆p+1 → X by

Dσ(t0, . . . , tp) ={h(σ(

t11−t0 , . . . ,

tp1−t0

), t0)

t0 6= 1x0 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∑i=1

(−1)i−1(Dσ)di = σ −p∑i=0

(−1)iD(σdi) = σ −D∂σ.

For p = 0 we get ∂Dσ − D∂σ = ∂Dσ = σ − σ0, where σ0 : ∆0 → X, 1 7→ x0. From thiswe conclude that D is really a chain homotopy as required and so HpX = 0 ∀p 6= 0 and forp = 0 our simplices σ and σ0 differ by a boundary, meaning that [σ] = [σ0] ∈ H0X and soH0X is an abelian group with one generator. �

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Paragraph 2. Homotopy Invariance 19

2. Homotopy InvarianceIn this section, we are going to prove the first of Eilenberg and Steenrod’s axioms for oursingular theory. So we must prove that for X a topological space ι0(X) and ι1(X) inducethe same morphisms in homology, where ιt(X) : X → X × I, x 7→ (x, t) for t ∈ {0, 1}. Aspromised in chapter 2, we are going to use the following models to apply the acyclic modeltheorem 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 acyclicmodel theorem to singular homology.

(2.2) Remark. The functor S : Top → AbGrp is free with models S and aacyclic onS. It is obviously aacyclic since all the ∆q are contractible. The freeness has to be checkedat each degree. To do so, let p ∈ N and we consider S ′ := {∆p} ⊂ S and the elementep := 1∆p ∈ Sp∆p. By definition for X any space, the set {σ∗ep = σ ∈ SpX | σ : ∆p → X}forms a basis for SpX.

(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 → Yto (f × 1I)∗ : S(X × I)→ S(Y × I). Since M × I is contractible for all M ∈ S this functoris aacyclic on S (cf. (1.3)). For F := S : Top → Ch(AbGrp), which is free with modelsS as seen above, we can apply the acyclic model theorem. Clearly ι0, ι1 : F → G aretwo natural transformations and they induce the same morphism H0X → H0(X × I). Tosee this, let σ : ∆0 → X, 1 7→ x0 be an arbitrary 0-simplex (i.e. a point) and considerb := (ι1(X))∗σ − (ι0(X))∗σ ∈ S0(X × I). Define c : ∆1 ∼= I → X × I, t 7→ (x0, t) and itfollows that ∂c = (x0, 1)− (x0, 0) and so ∂c = ι1(X) ◦ σ − ι0(X) ◦ σ meaning that (ι0(X))∗σand (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 singularsimplex

[v0, . . . , vp] : ∆p → ∆q,p∑i=0

tiei 7→p∑i=0

tivi

and dub ξp := [e0, . . . , ep] = 1∆p : ∆p → ∆p. One easily calculates that for p > 1

∂[v0, . . . , vp] =p∑i=0

(−1)i[v0, . . . , vi, . . . , vp].

Geometrically, [v0, . . . , vp](∆p) ⊂ ∆q ⊂ Rq+1 is the (possibly degenerate) p-simplex spannedby 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].

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20 Chapter 3. Singular Homology

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-chainsc ∈ Sp∆q consisting of affine singular simplices. Explicitly, if σj = [vj0, . . . , vjp], p > 0 is anaffine singular simplex and mj ∈ Z for j ∈ {1, . . . , n}, we define for c =

∑nj=1mjσj ∈ Sp∆q

the v-cone vc :=∑nj=1mjvσj ∈ 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 awholely trivial remark as some people put [ ] := 0 by applying the above law for calculatingthe boundary in the case p = 0 (so one has ∂[v0] = [ ] = 0) but then one is tempted tocalculate v[ ] = [v], which cannot be the case since [ ] = 0 and v is a homomorphism. To avoidsuch unnecessary confusion, we leave the expression [ ] undefined and will need an additionalcase differentiation here and there.

(3.2) Lemma. For v ∈ ∆q and σ = [v0, . . . , vp] : ∆p → ∆q an affine singular simplex wehave ∂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∑i=0

(−1)i[v, v0, . . . , vi, . . . , vp] = σ − v∂σ

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 ofσ and we look at the v-cone over the faces of σ. This will lead to the so-called barycentricsubdivision. Let σ = [v0, . . . , vp] : ∆p → ∆q be an affine singular simplex. We write bσ :=1/p+1

∑pi=0 vi for the barycentre of σ and as a special case, we write bp := bξp = 1/p+1

∑pi=0 ei.

We can now define the barycentric subdivision B recursively over p. To do this for anarbitrary singular simplex, we must first define the barycentric subdivision for the universalsimplices ξp

The geometric idea of barycentric subdivision is to divide a p-simplex into smallerand smaller simplices so that at some point these simplices are small enough as to fit entirelyinto an open subset of a given cover (i.e. their diameter is smaller than a Lebesgue numberof this cover) thus enabling us to use Lebesgue’s Lemma. For p = 0 ξp is just a point andwe cannot subdivide anything. For p = 1 the (image of the) simplex ξp is an interval and weidentify this with [0, 1]. We subdivide it in the two pieces [0, 1/2] and [1/2, 1], i.e. we connectthe endpoints to the barycentre. Assume now, that the barycentric subdivision is defined fordimension p− 1 then we define it for ξp by subdividing each face (which are p− 1-simplices)and taking the bp-cone, i.e. connecting each face with the barycentre. Pictorially (here forthe standard 2-simplex ξ2) this is just

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(3.3) Definition. (Barycentric Subdivision) For X a topological space and c ∈ S0Xa 0-chain we define the barycentric subdivision of c as BXc := c. Assume now that BX isdefined for all topological spaces X and all (p− 1)-chains in Sp−1X (with p > 1). To extendthis to p-chains, we first define the barycentric subdivision B∆pξ

p of the universal p-simplex

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Paragraph 3. Barycentric Subdivision 21

ξp = 1∆p by B∆pξp := bpB∆p(∂ξp) (i.e. subdivide the faces and connect with the barycentre).

For an arbitrary topological space X we define BX : SpX → SpX on the basis elements (i.e.the singular p-simplices) and extend this linearly to all p-chains. So if σ : ∆p → X is anysingular p-simplex we subdivide σ by writing σ = σ ◦ ξp and subdividing ξp. I.e. we defineBXσ := σ∗B∆pξ

p.

For the following theorem, in the case where X = ∆q for some q ∈ N, let us fix thenotation ApX to denote the subgroup of SpX spanned by the affine singular simplices. Anelement of ApX will be called an affine p-chain. By (3.2) the boundary maps ∂ map Ap+1Xto ApX, i.e. the image of an affine p + 1-chain is an affine p-chain. So the ApX togetherwith the singular chain maps form again a chain complex AX. Obviously, if f : X → X ′

is continuous and X ′ = ∆q′ for some q′ ∈ N then f∗ : SX → SX ′ restricts to a chainmorphism f∗ : AX → AX ′ and in particular the barycentric subdivision restricts to a mapBX : ApX → ApX by definition of BX .

(3.4) Theorem. For X a topological space BX : SX → SX is a natural chain map. Putdifferently, B is a natural endotransformation B : S → S. Explicitly this means

(a) BX : SpX → SpX is natural in X for each p ∈ Z. I.e. for f : X → Y in Top wehave f∗ ◦BX = BY ◦ f∗.

(b) If X = ∆q for some q ∈ N and σ = [v0, . . . , vp] : ∆p → X is an affine singularsimplex then BXσ = bσBX∂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 bynaturality. 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 SpX.

f∗BXσ = f∗σ∗ (bpB∆p∂ξp) and also BY f∗σ = (f ◦ σ)∗ (bpB∆p∂ξ

p) .

Ad (b): This follows easily from (a):

BXσ = σ∗B∆pξp = σ∗(bpB∆p∂ξ

p) = bσσ∗B∆p∂ξp (a)= bσBXσ∗∂ξ

p = bσBX∂σ.

Ad (c): We first prove our proposition for X = ∆q and affine singular simplices by inductionon p. The case p = 0 is trivial as BX : S0X → S0X is simply the identity map. For p = 1let σ : ∆1 → X be an affine singular 1-simplex. Then ∂σ is a 0-chain and so BX∂σ = ∂σ bydefinition. On the other hand

∂BXσ = ∂(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

∂BXσ = ∂(bσBX∂σ) (3.2)= BX∂σ − bσ∂BX∂σind. hyp.= BX∂σ − bσBX∂∂σ = BX∂σ.

We conclude that for X = ∆q we have BX ◦ ∂ = ∂ ◦ BX on AX. If X is now an arbitraryspace and σ : ∆p → X a singular simplex, we have

∂BXσ = ∂σ∗B∆pξp = σ∗∂B∆pξ

p = σ∗B∆p∂ξp = BXσ∗∂ξ

p = BX∂σ,

where we used in the third equality that ∂ ◦B∆p = B∆p ◦ ∂ on A∆p as shown above. �

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22 Chapter 3. Singular Homology

(3.5) Remark. If our topological space X is fixed and there is no risk of confusion, wewill usually omit the index and simply write B for BX .

(3.6) Remark. Each simplex in the p-chain B∆qσ is a subsimplex of σ (i.e. the image ofthese 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 applyLebesgue’s Lemma. I.e. for a given open cover we want to apply the barycentric subdi-vision 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 underbarycentric subdivision.

(3.7) Theorem. Let σ = [v0, . . . , vp] : ∆p → ∆q be an affine singular simplex. Then

(a) diam σ(∆p) = max {‖vi − vj‖ | i, j ∈ {0, . . . , p}};

(b) For all x ∈ σ(∆p) ‖bσ − x‖ 6 pp+1 diam σ(∆p);

(c) Every simplex in the chain Bσ = B∆qσ has diameter at most pp+1 diam σ∆p.

Proof. Ad (a): Clearly maxi,j ‖vi − vj‖ 6 diam σ(∆p) since vi ∈ σ(∆p) ∀i. So we have toprove the other inequality. For x ∈ Rq+1 and y =

∑pi=0 tivi ∈ σ(∆p) with

∑pi=0 ti = 1 and

ti > 0 ∀i we calculate

‖x− y‖ =∥∥∥∥∥p∑i=0

tix−p∑i=0

tivi

∥∥∥∥∥ 6p∑i=0

ti ‖x− vi‖ 6p∑i=0

ti maxj‖x− vj‖ = max

j‖x− vj‖ .

Putting x =∑pi=0 sivi ∈ σ(∆p), we get the desired result by a similar calculation.

Ad (b): By definition bσ = 1p+1

∑pi=0 vi. If x =

∑pj=0 tjvj ∈ σ(∆p) with

∑pj=0 tj = 1 and

tj > 0 ∀j it is easy to see that for all j ∈ {0, . . . , p}

‖bσ − vj‖ =∥∥∥∥∥ 1p+ 1

p∑i=0

vi − vj

∥∥∥∥∥ =∥∥∥∥∥ 1p+ 1

p∑i=0

vi −p+ 1p+ 1

vj

∥∥∥∥∥ =∥∥∥∥∥ 1p+ 1

p∑i=0

(vi − vj)∥∥∥∥∥

= 1p+ 1

∥∥∥∥∥p∑i=0

(vi − vj)∥∥∥∥∥ 6

1p+ 1

p∑i=0‖vi − vj‖ = 1

p+ 1

p∑i=0i 6=j

‖vi − vj‖

6p

p+ 1maxi‖vi − vj‖ .

And so

‖bσ − x‖ 6p∑j=0

tj ‖bσ − vj‖ 6p

p+ 1

p∑j=0

tj maxi,j‖vi − vj‖ = p

p+ 1diam σ(∆p).

Ad (c): For this proof, for the sake of readability, let us fix the notation mesh c for a p-chain c =

∑mi=0 niσi (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. Forp = 0, σ(∆p) ∈ ∆q is a point and so diam σ(∆p) = 0 and since Bσ = σ by definition theproposed inequality holds. For the inductive step, remember that Bσ = bσ(B∂σ) and by(3.6) meshBσ = max{ p

p+1 diam σ(∆p),mesh(B∂σ)} (since every simplex in ∂σ is containedin σ) and by the induction hypothesis mesh(B∂σ) 6 p−1

p mesh ∂σ 6 pp+1 mesh ∂σ. �

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Paragraph 4. Small Simplices and Standard Models 23

4. Small Simplices and Standard Models

(4.1) Definition. Let X be a topological space and U ⊂ PX a collection of subsets ofX such that the interiors of all U ∈ U cover X. A singular simplex σ : ∆p → X is calledU-small iff σ(∆p) ⊂ U for some U ∈ U . More generally a p-chain c =

∑mi=1 niσi is called

U-small iff all the σi are U-small. The subgroup of SpX spanned by the U-small simplicesgives us an abelian group SpU and this defines a subcomplex SU of SX (obviously the imageof 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’sLemma and conclude.

(4.2) Corollary. LetX = ∆q be a standard model and U ⊂PX a collection of subsets ofX, whose interiors form an open cover of X. For any singular simplex (σ : ∆p → X) ∈ SpX,there is a k ∈ N such that Bkσ ∈ SpU .

Proof. (σ−1U)U∈U is an open cover of ∆p and since this is compact there is a Lebesguenumber ε ∈ R>0 such that every V ⊂ X with diamV < ε is wholely contained in σ−1U forsome U ∈ U . By (3.7) the proposition follows. �

(4.3) Remark. We have already seen in (2.2) that the functor S : Top → AbGrp isfree with models S and aacyclic on S. So, we can use corollary (2.4) of the last chapter toconclude 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 omitthe index if there is no risk of confusion). By the naturality of D it follows that DX maps SUinto itself for any cover U of X. To see this, let σ : ∆p → X be a U-small singular p-simplex,i.e. σ factors as ∆p σ−→ U

i−→ X for some U ∈ U , where i : U ↪→ X is the inclusion. Sincei∗ ◦DU = DX ◦ i∗ (this is the naturality of D) it follows that DXσ = DX ◦ i∗σ = i∗ ◦DUσand because i∗ : SU ↪→ SX is again the inclusion DXσ ∈ 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 andisomorphism i∗ : H(SU) ∼= H(SX).

Proof. Clearly H0(SU) = S0U = S0X = H0(SX) and we have to prove that i∗ : Hp(SU) →Hp(SX) is an isomorphism for all p > 1. For c ∈ ZpX = ker(∂ : SpX → Sp−1X) a p-cycle bythe last corollary (4.2) there is a k ∈ N such that Bkc ∈ SpU . Then Bkc−c = Dk∂c+∂Dkc =∂Dkc and so [c] = [Bkc] ∈ HpX, which proves the surjectivity of i∗. For injectivity, we mustprove that Bp(SX) = im(∂ : Sp+1X → SpX) ⊂ Bp(SU). I.e. we show that each p-chainof SU , which is a boundary in SX is already a boundary in SU . So let c = ∂b′ ∈ SpUfor some b′ ∈ Sp+1X. By the last corollary (4.2) there is a k ∈ N such that Bkb′ ∈ Sp+1Uand by the above remark Dkc ∈ Sp+1U and we put b := Bkb′ − Dkc ∈ Sp+1U and becauseBk − 1SX = ∂Dk +Dk∂, obviously

∂b = ∂Bkb′ − ∂Dkc = Bk∂b′ − (Bkc− c−Dk∂c) = Bkc−Bkc+ c+Dk∂∂b′ = c. �

With this lemma at hand we can finally embark on the task of proving the excisionaxiom for singular homology.

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24 Chapter 3. Singular Homology

5. Excision

This section is solely devoted to proving the excision axiom for singular homology. Here ourinvestment into the machinery of homological algebra (in the form of the acyclic model theo-rem) pays off again and the proof boils down to simple algebra with no geometric argumentswhatsoever.

(5.1) Definition. We define the category Cov having as objects all pairs (X,U) whereX is a topological space and U ⊂ PX is a collection of subsets of X, whose interiorsU :=

{U∣∣∣ U ∈ U} cover X. An arrow f : (X,U)→ (Y,V) consists of a map f : X → Y such

that U = f−1V :={f−1V

∣∣ V ∈ V}. We equip this category with the models

M :={(∆q,U)

∣∣∣ q ∈ N, U ⊂PX arbitrary, such that U covers X}.

We define two functors S′, S′′ : Cov → Ch(AbGrp) on the objects by S′(X,U) := SX,S′′(X,U) := SU and and the arrows by S′f = S′′f = f∗.

We are now able to prove our main theorem. The proof is essentially the same as theproof 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′ and S′′ are both free with modelsM and aacyclic onM.

Proof. We first check the freeness, which we have to check at each degree p ∈ N. We firstchoose

Mp :={(∆p,U)

∣∣∣ U ⊂PX, such that U covers ∆p}⊂M

and ep := 1S∆p ∈ S′(∆p,U) = S∆p for (∆p,U) ∈ Mp. For an arbitrary object (X,U) inCov and σ : ∆p → X and arbitrary singular p-simplex in SpX there is one and only onecover, namely V := σ−1U , of ∆p such that σ defines an arrow σ : (∆p,V) → (X,U). Soσ ∈ Cov ((∆p,V), (X,U)) and

{σ∗ep = σ | (∆p,V) ∈Mp, σ ∈ Cov ((∆p,V), (X,U))}

is a basis for SpX and so S′ is free with modelsM. By the same argument (with ep := 1SUinstead of 1S∆p) S′′ is free with modelsM.

Clearly S′ is aacyclic on M and so is S′′ by the lemma (4.4) above, which provesour proposition. �

Again by the above lemma (4.4) there is a natural isomorphism ϕ : H0S′′ ∼= H0S

defined by ϕ(X,U) := i∗ : H0(SU) → H0(SX), where i : SU ↪→ SX is the inclusion. So bycorollary (2.3) in chapter 2 this can be lifted to a natural isomorphism ϕ : S′′ ∼= S′ and as animmediate consequence we get the following corollary.

(5.3) Corollary. For X a topological space and U ⊂ PX such that U covers X, theinclusion SU ↪→ SX is a chain equivalence. �

(5.4) Corollary. (Excision for Singular Homology) Let (X,A) ∈ Ob(Top(2)) be apair, U ⊂ X open with U ⊂ A. 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.

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Paragraph 5. Excision 25

Proof. Consider U := {A,X \U} which satisfies the condition that U covers X. By definitionSU = SA+S(X \U) and the inclusion SU = SA+S(X \U) ↪→ SX is a chain equivalence bythe 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 anisomorphism

S(X \ U)/S(A \ U) ∼= SU/SA = (SA+ S(X \ U))/SA.

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. �

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LITERATURE

Dold, A.: Lectures on Algebraic Topology, Springer, Berlin; Heidelberg; New York, 1980

Eilenberg, S.; Steenrod, N.: Foundations of Algebraic Topology, Princeton UniversityPress, 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 Cen-tury, 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, NewYork, 1994

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INDEX

Numbers

1, 4

A

Aacyclic chain complex, 14Aacyclic functor

on models, 14Absolute homology, 5Absolute singular homology, 18Acyclic Model Theorem, 14Admissible

category for homology theory, 4Affine singular simplex, 19Almost acyclic functor

on models, 14

B

Barycentre, 20Barycentric subdivision, 20Boundary, 17Boundary morphism, 17Boundary operator, 5Braid diagram, 8Braid Lemma, 8

C

Categoryadmissible for homology theory, 4

Chain, 17affine, 21U-small, 23

Chain complexaacyclic, 14

Chain homotopicnatural transformations, 14

Coefficientsof a homology theory, 6

Cone, 19Connecting morphism, 5

D

Dimensioninvariance of, 12

E

Excision, 6for singular homology, 24

F

Faceof a singular simplex, 17

Free functorwith models, 13

H

Homologyof a pair, 5of a space, 5of spheres, 12reduced, 9

Homology sequence, 6of a triple, 9reduced, 10

Homology theory, 5Homotopic

maps in Top(2), 5natural transformations, 14

Homotopyin Top(2), 5

Homotopy Invariance, 5Homotopy invariance

of singular homology, 18

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28 Index

I

I, 4Inclusion, 3Invariance

of dimension, 12

L

Latticeof a pair, 4

Long Exact Homology Sequence, 6of a triple, 9reduced, 10

M

Map, 3Models

of a category, 13

N

Naturally chain homotopic, 14Naturally homotopic, 14

O

Ordinaryhomology theory, 6

R

Reduced homology, 9Relative homology, 5

of discs, 12Relative singular homology, 18Restriction functor, 5

S

Shift functor, 14Simplex, 17

affine singular, 19standard, 17U-small, 23

Singular homology, 18absolute, 18

relative, 18Singular p-chain, 17Singular simplex, 17Space, 3Standard models, 19Standard simplex, 17Subdivision

barycentric, 20

T

Top, 3Top(2), 3Top(3), 3

U

U-small chain, 23U-small simplex, 23

V

v-cone, 19