Homology Theory Kay Werndli 13

Home , Acyclic model, Reduced homology, Singular homology

Homology Theory Kay 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 found under http://creativecommons.org/licenses/by-nc-sa/2.5/ch/deed.en_GB. CONTENTS

Chapter 1 Axiomatic Homology Theory ...... 3 1. The Eilenberg-Steenrod Axioms...... 3 2. First Consequences...... 6 3. Reduced Homology...... 9 4. Homology of Spheres ...... 10 Chapter 2 Acyclic Models ...... 13 1. Models...... 13 2. The Acyclic Model Theorem ...... 14 Chapter 3 Singular Homology ...... 17 1. Deﬁnitions...... 17 2. Homotopy Invariance ...... 19 3. Barycentric Subdivision ...... 19 4. Small Simplices and Standard Models ...... 23 5. Excision...... 24 Literature ...... 26

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 fA ⊂ 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,B0) is a continuous map f : X → Y with 0 0 fA ⊂ 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 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 4 Chapter 1. Axiomatic Homology Theory

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

U / F / Top(2) o _ Top and Top o _ Top(2) , G 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) Deﬁnition. A subcategory C 6 Top(2) is called admissible for homology theory iﬀ

(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 1 =∼ {∗}, 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, ∅) u: JJ uu JJ uu JJ uu JJ uu J$ (∅, ∅) / (A, ∅) (X,A) / (X,X) . II t: II tt II tt II tt I$ tt (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

ι 0 / , (X,A) / (X × I,A × I) ι1

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 f0x = f(x, 0) and f1x = 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

ι 0 / f , (X,A) / (X × I,A × I) / (Y,B) ι1 satisfying f ◦ ι0 = f0 and f ◦ ι1 = f1.

(1.6) Definition. For C an admissible category, we define the so-called restriction functor A ρ : C → C which sends (X,A) to (A, ∅) and f :(X,A) → (Y,B) to ρf =: f|B :(A, ∅) → (B, ∅), x 7→ fx. 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∈ , where A is an abelian category and a family of natural trans- Z th formations (∂n : Hn → Hn−1 ◦ ρ) . Hn(X,A) is called the n homology of (X,A) and ∂n n∈Z the nth boundary operator or connecting morphism. As mentioned before, we identify X with th (X, ∅) and in the same spirit write HnX or Hn(X) for Hn(X, ∅), which we call the n (absolute) homology of X. To avoid unnecessarily complicated notation, we write f∗ for Hnf : 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) Hn(X,A) / Hn−1A

f f∗ 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)∗. 6 Chapter 1. Axiomatic Homology Theory

(ii) (Long Exact Homology Sequence) For each (X,A) ∈ Ob(C) we have a long exact 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 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 −10 ⊂ 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 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 coeﬃcient H01 is of any interest. If we have chosen ∼ an isomorphism H01 = 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∗ : HnA → HnX 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∗ : HnA → HnX 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.

A (2.2) Remark. If f :(X,A) → (Y,B) is such that f : X → A and f|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,

(i1)∗ (i2)∗ Hn(X1,A1) ⊕ Hn(X2,A2) / Hn(X,A) is an isomorphism. Proof. Consider the morphism (i1)∗ from the direct sum of the long exact homology se- (i2)∗ quences 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

i1 j1 i2 j2 X1 −→ X −→ (X,X1) and X2 −→ X −→ (X,X2), whose induced morphisms can be combined in a commutative diagram

f1 HnX1 / Hn(X,X2) HH 8 HH rrr HH rrr (i1)∗ HH r (j2)∗ H$ rrr HnX v: LL (i2)∗ vv LL (j1)∗ vv LLL vv LL vv L& HnX2 / Hn(X,X1) . 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. 8 Chapter 1. Axiomatic Homology Theory

(2.4) Lemma. Given a commutative diagram

f A 1 / B 1 I : 2 II uu II uu i1 I$ uu j2 : X I i2 uu IIj1 uu II uu I$ A2 / B1 f2 in an abelian category with exact diagonals. Then the following conditions are equivalent:

(a) f1 and f2 are isomorphisms; (b) i1 : A ⊕ A → X is an isomorphism; i2 1 2

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) −→ HnA → 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) qqMMM qqMMM qqMMM qq∂ MM qq∂ MM qq MM MM qq MMM qq MM qq MMM MM& qqq M& qqq MM& qqq & H (X,A) H (A, B) Hn−1B n8 +1 8 n 8 MM ∂ MM ∂ q MM 8 (2) qqq MM qqq MM qq MM qq q MM& qqq M& qq MM& qq H A H (X,B) H A 8 n n 8 n−1 ∂ q MM 8 M ∂ q MM (3) MM qq MM qqq MMM qq MM& MM& qqq MM& qqq M& qq HnB HnX Hn(X,A) ∂ 8 M 8 M 8 qq MM qq MM qq MMM 8 qq MM qq MM q M ∂ qq MM qq MM qqq MMM qqq (4) MMqqq MMqqq MMqqq , 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 Hn1 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

H˜nX := ker (p∗ : HnX → Hn1) .

For f : X → Y , we get again an induced morphism f∗ : HñX → HñY in the obvious way. Like that we can extend Hñ 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+1(X, x) −→ Hn1 −→ HnX → Hn(X, x) −→ Hn−11 −→ ....

Because p∗ ◦ x∗ = 1Hn1 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∗.

x∗ 0 / Hn1 / HnX / Hn(X, x) / 0 G GG GG p∗ 1 GG Hn1 GG# Hn1 .

i ˜ ∼ The triangle on the left gives us an isomorphism x∗ : HnX ⊕ Hn1 −→ HnX, where i : HñX,→ HnX is the standard inclusion. One plainly checks this as follows: ∼ ∼ HnX = ker p∗ ⊕ x∗(Hn1) = HñX ⊕ x∗(Hn1) = HñX ⊕ Hn1. By the first isomorphism theorem j | : H˜ X −→∼ H (X, x) is an isomorphism, where ∗ HñX n n j : X → (X, x) is the standard inclusion: ∼ ∼ ∼ Hn(X, x) = HnX/x∗(Hn1) = HñX ⊕ x∗(Hn1) /x∗(Hn1) = HñX and finally these two together give

i j | ×1 ∼ x∗ ∗ H˜nX Hn1 HnX = Hn(X, x)⊕Hn1 by HnX ←−−− H˜nX⊕Hn1 −−−−−−−−−→ Hn(X, x)⊕Hn1, 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 exact from which one easily deduces the following theorem.

(3.2) Theorem. If X is contractible, then H˜ X ∼ H (X, x) = 0 ∀n ∈ . n = n Z 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) → HnA lies in HnA. 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) −→ H˜nA → H˜nX → 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) / H˜nA / H˜nX / Hn(X,A) /

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

p∗ p∗ p∗ p∗ ... / H (1, 1) / H 1 / H 1 / H (1, 1) / ... 0 n+1 0 n ∼ n 0 n 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 Hn1 → 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 (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 n n+1 from the axioms. To do so, we observe ﬁrst, that we can divide the sphere S ⊂ R in an upper and lower hemisphere

n n D± := {(x1, . . . , xn+1) ∈ S | ±xn+1 > 0} . Paragraph 4. Homology of Spheres 11

n ∼ n n n n+1 Obviously D± = D by simply projecting D± along the xn+1-axis to R × {0} ⊂ R . 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 get 0 1 2 n th S ,→ S ,→ S ,→ ... . Let’s furthermore ﬁx the notation ei ∈ R to denote the i standard basis vector having (ei)j = δi,j (the Kronecker delta) and with this, let’s write N := en+1 n and S := −en+1 for the north and south pole of S respectively. Now, for n ∈ N>0 we look at the commutative diagram

n n−1 n n Hk(D−,S ) / Hk(S ,D+)

n n n n Hk(D−,D− \{S}) / Hk(S ,S \{S})

n−1 n n n induced by inclusions. Because S ,→ D \{S} and D+ ,→ S \{S} are homotopy equivalences, it follows that the vertical arrows are isomorphisms (by remark (2.2)). For the ˚n bottom arrow, we can use the excision axiom with U := D+ and deduce that this is also an isomorphism. In conclusion, the top arrow has to be an isomorphism, too. n−1 n We choose ∗ := e1 = (1, 0,..., 0) ∈ S ⊂ S and insert this isomorphism in a second diagram

n n−1 ∂ n−1 ∼ ˜ n−1 Hk(D−,S ) / Hk−1(S , ∗) = Hk−1S

o σ+ σ+ n n n n Hk(S ,D ) o H (S , ∗) =∼ ˜ + j k HkS

n n−1 The reduced long exact homology sequence of (D−,S ) reads as

˜ n n n−1 ∂ ˜ n−1 ˜ n ... → Hk(D−) → Hk(D−,S ) −→ Hk−1(S ) → Hk−1(D−) → ....

n ∼ n−1 ˜ n ˜ n Since D− = D is contractible, we deduce from theorem (3.2) that Hk(D−) = Hk−1(D−) = n 0 and so ∂ is an isomorphism. By the same argument, j is an isomorphism, since {∗} ,→ D− is a homotopy equivalence. Now we can easily deﬁne σ+ to be the unique isomorphism making the diagram commute.

˜ 0 ∼ (4.1) Lemma. HnS = Hn1 for all n ∈ Z.

Proof. Choose a point x : 1 → S0 and denote the other point by y : 1 → S0. Let’s also 0 0 write i : H˜nS ,→ HnS for the standard inclusion. As seen in the section about the reduced homology and theorem (2.3), we get a commutative diagram

i y∗ x ( x ) 0 ∗ 0 ∗ H˜ S ⊕ H 1 / H S o Hn1 ⊕ Hn1 n O n ∼ n ∼ O

O O , Hn1 Hn1 12 Chapter 1. Axiomatic Homology Theory where the vertical arrows are the standard inclusions into the second summand. If we deﬁne y∗ −1 i ˜ 0 ∼ the isomorphism f := ( x∗ ) ◦ x∗ : HnS ⊕ Hn1 −→ Hn1 ⊕ Hn1, we can rewrite this as a commutative diagram

0 0 0 / Hn1 / / H˜nS ⊕ Hn1 / / H˜nS / 0

o f o fˆ , 0 / Hn1 / / Hn1 ⊕ Hn1 / / Hn1 / 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 n ∼ n ∼ H˜kS = Hk−n1 and HkS = Hk−n1 ⊕ Hk1. It follows that we also have isomorphisms n+1 n ∼ ˜ n ∼ Hk(D ,S ) = Hk−1S = Hk−(n+1)1.

Proof. As seen in the last paragraph, we have isomorphisms n ∼ n−1 ∼ ∼ 0 ∼ H˜kS = H˜k−1S = ... = H˜k−nS = Hk−n1, where we used the above lemma for the last isomorphism. By remembering ourselves that n ∼ n HkS = H˜kS ⊕ 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

n+1 n ∂ n 0 → Hk(D ,S ) −→ H˜k−1S → 0 and so ∂ is an isomorphism.

(4.3) Corollary. Let (Hn)n∈ , (∂n)n∈ be an ordinary homology theory, having coeﬃ- ∼ Z Z cient H01 = G then for n ∈ N>0 ( ( G k ∈ {0, n} G k = n + 1 H Sn =∼ and similarly H (Dn+1,Sn) =∼ k 0 otherwise k 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

m ∼ n (4.4) Corollary. (Invariance of Dimension) R = R ⇔ m = n for m, n ∈ N. Proof. The direction “⇐” is trivial and for the other direction, we assume that for m 6= n we m n we have a homeomorphism R → R . The case where m = 0 or n = 0 is trivial an so the m n case m, n > 1 is left. We can extend our homeomorphism R → R to a homeomorphism of the one-point compactiﬁcations, which is Sm =∼ Sn but since m 6= n by the above theorem H Sm = 0 but H Sn ∼ G 6= 0. n n = (4.5) Corollary. Sn 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 q q+1 “models” of this theory in a topological context are the standard simplices ∆ ⊂ R . And q in fact, we will investigate {∆ | 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 0 0 models M iﬀ there is a subset M ⊂ M and for each M ∈ M an element eM ∈ FM such that for every C ∈ Ob(C) the module FC is free and the set

0 (F a)eM ∈ FC M ∈ M , a ∈ C(M,C) forms a basis for FC. 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 :(FC)p → (FD)p. One should notice that at 14 Chapter 2. Acyclic Models

p each degree, we can have a different subset Mp ⊂ M and different elements eM ∈ FnM 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 FM is exact everywhere except at the degree 0. I.e. Hi(FM) = 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 : 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 is required to be natural in C. That is for each a : C → D in C the following diagram commutes

χ C FC C / GC

a F a Ga DFD / GD . χD 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 X0 = S−X, 0 − having Xn = 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 : H0F → H0G (where H0F,H0G : C → R-Mod) then there is a natural transformation ϕ˜ : 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 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 ϕ˜ as to make the diagram

∂ ∂ ∂ ∂ ... / Fn+1C / FnC / ... / F0C / / H0(FC) / 0

ϕ˜ ϕ˜ ϕ˜ ϕ ... / Gn+1C / G C / ... / G C / / H0(GC) / 0 ∂ ∂ n ∂ ∂ 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” ϕ˜ 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 n an element eM ∈ FnM as in the deﬁnition of a free functor with models M. Now, let C ∈ Ob(C) be arbitrary. Again by presumption, F0C = Z0(FC) and G0C = Z0(GC) are the cycles of FC and GC at degree 0 respectively. Thus, we get standard projections onto the 0th homology modules as in the following diagram

F0C / / H0(FC)

ϕ˜ 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 in degree −1, where it is simply ϕ. For the inductive step let’s assume that ϕ˜ is defined in degree n − 1 for n > 0. For n each model M ∈ Mn we consider ϕ˜(∂eM ) ∈ Gn−1M (which we have already defined). Since the diagram

∂ ∂ FnM / Fn−1M / Fn−2M

ϕ˜ ϕ˜ ϕ˜ G M / Gn−1M / Gn−2M . n ∂ ∂ commutes and the lower row is exact (because G is aacyclic on M) we conclude that n n n ∂ϕ˜(∂eM ) =ϕ ˜(∂∂eM ) = 0 and so ϕ˜(∂eM ) ∈ Gn−1M must be a boundary. I.e. we can n n choose c ∈ GnM satisfying ∂c =ϕ ˜(∂eM ) and define ϕe˜ M := c, which makes the above diagram commute. n For a : M → C a morphism in C (Fna)eM ∈ FnC is a basis element of FnC and we n n define ϕ˜ ((Fna)eM ) := (Gna)ϕe ˜ M . We do this for every a and every M and like that define ϕ˜ on 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), we look at the following cubical diagram

ϕ˜ FnM / GnM Fna rrr Gna rrr rrr rrr ry r ϕ˜ ry r FnC / GnC

Fn−1M / Gn−1M ϕ˜ r r rrr rrr y rrr Fn−1a y rrr Gn−1a r r , Fn−1C / Gn−1C ϕ˜ 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 n eM ∈ FnM the top and the back faces commute by deﬁnition of ϕ˜. So the front face has 16 Chapter 2. Acyclic Models

n to commute, too for (Fna)eM ∈ FnC. Since this holds for all M and all a the front face commutes for all basis elements of FnC and so commutes as a whole. This defines ϕ˜ in degree n. One easily checks the naturality of ϕ˜, i.e. for a : C → D an arrow in C the equation ϕ˜ ◦ F a = Ga ◦ ϕ˜ holds. What is left to prove is the uniqueness up to natural homotopy. So suppose that ϕ,˜ 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 chain homotopy 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 with the augmented F and G), where it is simply 0 because there ϕ˜ = ψ˜ = ϕ. Suppose now, that n χ is defined in degree n − 1 with n > 0. FnC has {(F a)eM }M,a as a basis and we notice that n ˜ n n b :=ϕe ˜ M − ψeM − χ(∂eM ) is a cycle because

n ˜ n n n ˜ n n n ˜ n ∂ϕe˜ M − ∂ψeM − ∂χ∂(eM ) =ϕ ˜(∂eM ) − ψ(∂eM ) − −χ(∂∂eM ) +ϕ ˜(∂eM ) − ψ(∂eM ) = 0.

Because GM is aacyclic, b must be a boundary, i.e. there is a c ∈ Gn+1M with ∂c = b and n n n we define χeM := c. By defining χ ((Fna)eM ) := (Gn+1a)χeM we have defined χ on all basis elements and thence can extend it uniquely to χ : FnC → Gn+1C. 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 :ϕ ˜ ' ψ˜ 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 : H0F = H0G, then ϕ 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 with models M and aacyclic on M and α : F → F is a natural endotransformation inducing th the identity in 0 homology. Then α is naturally homotopic to 1F . In particular, for each C ∈ 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 which induces 1H0F : H0F → H0F and is unique up to homotopy. But α and 1F are two such natural transformations and so α and 1 are naturally homotopic. F 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.

p+1 (1.1) Definition. For p ∈ N let ei ∈ R be the i-th standard basis vector. We define the p-dimensional standard simplex as n o p p+1 ∆ := (t0, . . . , tp) = t0e0 + ... + tpep ∈ R t0 + ... + tp = 1 and ti > 0 ∀i . Every function α : {0, . . . , p} → {0, . . . , q} induces an affine map

p p p q X X ∆α : ∆ → ∆ , tiei 7→ tieαi. i=0 i=0 p Especially interesting for our theory is the injective map δi : {0, . . . , p − 1} → {0, . . . , p} for p p i ∈ {0, . . . , p} which simply leaves out the value i. We define di := ∆δi and will usually omit the upper index where it is unnecessary. For X a topological space, we call a map p p th σ : ∆ → X a singular p-simplex in X and call σ ◦ di its i 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∗ : SpX → SpY which is defined on the basis elements of SpX by f∗σ := f ◦ σ. p An element of SpX is called a singular p-chain. The faces of a singular simplex σ : ∆ → X can be used to define a morphism p X i p ∂p : SpX → Sp−1X, σ 7→ (−1) σ ◦ di i=0 th and again, we omit the indices where they are clear from the context. We call ∂p the p boundary morphism or simply the pth boundary. One easily checks that

p+1 p p+1 p δj ◦ δi = δi ◦ δj−1 ∀p ∈ N, i ∈ {0, . . . , p}, j ∈ {0, . . . , p + 1}, i < j 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 th to f∗ : SpX → SpY in each degree. We write HpX := Hp(SX) and call this the p 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 fA ⊂ B. th As usual, we call Hp(X,A) := Hp(S(X,A)) the p 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

... −→∂ H A → H X → H (X,A) −→∂ H A → ... → H (X,A) → 0. p p p p−1 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. 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 constant map. We deﬁne a chain homotopy D : 1SX ' 0, i.e. a chain map of degree 1 satisfying p ∂D + D∂ = 1SX . To do so, we consider a singular simplex σ : ∆ → X (i.e. a basis element p+1 of SpX) and deﬁne Dσ : ∆ → X by ( t1 tp h σ 1−t ,..., 1−t , t0 t0 6= 1 Dσ(t0, . . . , tp) = 0 0 x0 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

p+1 p X i−1 X i ∂Dσ = (Dσ)d0 − (−1) (Dσ)di = σ − (−1) D(σdi) = σ − D∂σ. i=1 i=0

0 For p = 0 we get ∂Dσ − D∂σ = ∂Dσ = σ − σ0, where σ0 : ∆ → X, 1 7→ x0. From this we conclude that D is really a chain homotopy as required and so HpX = 0 ∀p 6= 0 and for p = 0 our simplices σ and σ0 diﬀer by a boundary, meaning that [σ] = [σ0] ∈ H0X and so H X is an abelian group with one generator. 0 Paragraph 2. Homotopy Invariance 19

2. 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 chapter2, 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

q S := {∆ | 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 0 p at each degree. To do so, let p ∈ N and we consider S := {∆ } ⊂ S and the element p p ep := 1∆p ∈ Sp∆ . By deﬁnition for X any space, the set {σ∗ep = σ ∈ SpX | σ : ∆ → 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 → 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 H0X → H0(X × I). To 0 see this, let σ : ∆ → X, 1 7→ x0 be an arbitrary 0-simplex (i.e. a point) and consider 1 ∼ b := (ι1(X))∗σ − (ι0(X))∗σ ∈ S0(X × I). Deﬁne c : ∆ = 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 (ι (X)) σ diﬀer only by a boundary and so are the same in homology. 1 ∗

3. Barycentric Subdivision

q (3.1) Definition. Let v0, . . . , vp ∈ ∆ . We define [v0, . . . , vp] to be the affine singular simplex p p p q X X [v0, . . . , vp] : ∆ → ∆ , tiei 7→ tivi i=0 i=0 p p p and dub ξ := [e0, . . . , ep] = 1∆p : ∆ → ∆ . One easily calculates that for p > 1 p X i ∂[v0, . . . , vp] = (−1) [v0,..., vî, . . . , vp]. i=0 p q q+1 Geometrically, [v0, . . . , vp](∆ ) ⊂ ∆ ⊂ R is the (possibly degenerate) p-simplex spanned q by v0, . . . , vp. For v ∈ ∆ and p > 0, we define the v-cone over σ = [v0, . . . , vp] as

vσ = v[v0, . . . , vp] := [v, v0, . . . , vp]. 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-chains q j j c ∈ Sp∆ consisting of affine singular simplices. Explicitly, if σj = [v0, . . . , vp], p > 0 is an Pn q affine singular simplex and mj ∈ Z for j ∈ {1, . . . , n}, we define for c = j=1 mjσj ∈ Sp∆ Pn p the v-cone vc := j=1 mjvσj ∈ Sq+1∆ . As always for homomorphisms, we define v0 = 0 q q and so in particular v : Sp∆ → Sp+1∆ 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.

q p q (3.2) Lemma. For v ∈ ∆ and σ = [v0, . . . , vp] : ∆ → ∆ 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 p X i ∂vσ = ∂[v, v0, . . . , vp] = [v0, . . . , vp] − (−1) [v, v0,..., vˆi, . . . , vp] = σ − v∂σ i=0 and similarly for p = 0

∂vσ = ∂[v, v ] = [v ] − [v] = σ − [v]. 0 0

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 p q subdivision. Let σ = [v0, . . . , vp] : ∆ → ∆ be an affine singular simplex. We write bσ := 1 Pp 1 Pp /p+1 i=0 vi for the barycentre of σ and as a special case, we write bp := bξp = /p+1 i=0 ei. 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 ξp 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 ξp is just a point and we cannot subdivide anything. For p = 1 the (image of the) simplex ξp 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 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 for the standard 2-simplex ξ2) this is just

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(3.3) Definition. (Barycentric Subdivision) For X a topological space and c ∈ S0X 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−1X (with p > 1). To extend p this to p-chains, we first define the barycentric subdivision B∆p ξ of the universal p-simplex Paragraph 3. Barycentric Subdivision 21

p p p ξ = 1∆p by B∆p ξ := bpB∆p (∂ξ ) (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 any singular p-simplex we subdivide σ by writing σ = σ ◦ ξp and subdividing ξp. I.e. we deﬁne p BX σ := σ∗B∆p ξ .

q For the following theorem, in the case where X = ∆ for some q ∈ N, let us fix the notation ApX to denote the subgroup of SpX spanned by the affine singular simplices. An element of ApX will be called an affine p-chain. By (3.2) the boundary maps ∂ map Ap+1X to ApX, i.e. the image of an affine p + 1-chain is an affine p-chain. So the ApX together with the singular chain maps form again a chain complex AX. Obviously, if f : X → X0 0 q0 0 0 is continuous and X = ∆ for some q ∈ N then f∗ : SX → SX restricts to a chain 0 morphism f∗ : AX → AX and in particular the barycentric subdivision restricts to a map BX : ApX → ApX 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 : SpX → SpX is natural in X for each p ∈ Z. I.e. for f : X → Y in Top we have f∗ ◦ BX = BY ◦ f∗.

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

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∗. p We can now check the naturality of B on a basis element σ : ∆ → X in SpX.

p p f∗BX σ = f∗σ∗ (bpB∆p ∂ξ ) and also BY f∗σ = (f ◦ σ)∗ (bpB∆p ∂ξ ) . Ad (b): This follows easily from (a):

p p p (a) p BX σ = σ∗B∆p ξ = σ∗(bpB∆p ∂ξ ) = bσσ∗B∆p ∂ξ = bσBX σ∗∂ξ = bσBX ∂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 : S0X → S0X is simply the identity map. For p = 1 1 let σ : ∆ → 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.

p For p > 2 we can apply (3.2) and calculate for an aﬃne singular simplex σ : ∆ → X

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

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

p p p p ∂BX σ = ∂σ∗B∆p ξ = σ∗∂B∆p ξ = σ∗B∆p ∂ξ = BX σ∗∂ξ = BX ∂σ,

p where we used in the third equality that ∂ ◦ B p = B p ◦ ∂ on A∆ as shown above. ∆ ∆ 22 Chapter 3. Singular Homology

(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 q p these simplices in ∆ is contained in σ(∆ )). 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.

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

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

p p p (b) For all x ∈ σ(∆ ) kbσ − xk 6 p+1 diam σ(∆ );

p p (c) Every simplex in the chain Bσ = B∆q σ has diameter at most p+1 diam σ∆ . p p Proof. Ad (a): Clearly maxi,j kvi − vjk 6 diam σ(∆ ) since vi ∈ σ(∆ ) ∀i. So we have to q+1 Pp p Pp prove the other inequality. For x ∈ R and y = i=0 tivi ∈ σ(∆ ) with i=0 ti = 1 and ti > 0 ∀i we calculate p p p p X X X X kx − yk = tix − tivi ti kx − vik ti max kx − vjk = max kx − vjk . 6 6 j j i=0 i=0 i=0 i=0 Pp p Putting x = i=0 sivi ∈ σ(∆ ), we get the desired result by a similar calculation. 1 Pp Pp p Pp Ad (b): By deﬁnition bσ = p+1 i=0 vi. If x = j=0 tjvj ∈ σ(∆ ) with j=0 tj = 1 and tj > 0 ∀j it is easy to see that for all j ∈ {0, . . . , p}

p p p 1 1 p + 1 1 kb − v k = X v − v = X v − v = X(v − v ) σ j p + 1 i j p + 1 i p + 1 j p + 1 i j i=0 i=0 i=0 p p p 1 1 1 = X(v − v ) X kv − v k = X kv − v k p + 1 i j 6 p + 1 i j p + 1 i j i=0 i=0 i=0 i6=j p 6 max kvi − vjk . p + 1 i

And so p p X p X p p kbσ − xk 6 tj kbσ − vjk 6 tj max kvi − vjk = diam σ(∆ ). p + 1 i,j p + 1 j=0 j=0

Ad (c): For this proof, for the sake of readability, let us ﬁx the notation mesh c for a p- Pm chain c = i=0 niσi (with ni 6= 0 ∀i) to denote the maximum diameter of the σi. I.e. p mesh c := max {diam σi(∆ ) | 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 p (3.6) mesh Bσ = max{ p+1 diam σ(∆ ), mesh(B∂σ)} (since every simplex in ∂σ is contained in σ) and by the induction hypothesis mesh(B∂σ) p−1 mesh ∂σ p mesh ∂σ. 6 p 6 p+1 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 of X such that the interiors of all U ∈ U cover X. A singular simplex σ : ∆p → X is called p Pm U-small iff σ(∆ ) ⊂ U for some U ∈ U. More generally a p-chain c = i=1 niσi is called U-small iff all the σi are U-small. The subgroup of SpX spanned by the U-small simplices gives us an abelian group SpU 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 p X, whose interiors form an open cover of X. For any singular simplex (σ : ∆ → X) ∈ SpX, k there is a k ∈ N such that B σ ∈ SpU.

−1 p Proof. (σ U˚)U∈U is an open cover of ∆ and since this is compact there is a Lebesgue −1 number ε ∈ R>0 such that every V ⊂ X with diam V < ε is wholely contained in σ 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, i.e. σ factors as ∆p −→σ U −→i 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) = 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 by k k k k the last corollary (4.2) there is a k ∈ N such that B c ∈ SpU. Then B c−c = D ∂c+∂D c = k k ∂D c and so [c] = [B c] ∈ HpX, which proves the surjectivity of i∗. For injectivity, we must prove that Bp(SX) = im(∂ : Sp+1X → SpX) ⊂ Bp(SU). I.e. we show that each p-chain 0 of SU, which is a boundary in SX is already a boundary in SU. So let c = ∂b ∈ SpU 0 k 0 for some b ∈ Sp+1X. By the last corollary (4.2) there is a k ∈ N such that B b ∈ Sp+1U k k 0 k and by the above remark D c ∈ Sp+1U and we put b := B b − D c ∈ Sp+1U and because k k k B − 1SX = ∂D + D ∂, obviously

∂b = ∂Bkb0 − ∂Dkc = Bk∂b0 − (Bkc − c − Dk∂c) = Bkc − Bkc + c + Dk∂∂b0 = c.

With this lemma at hand we can ﬁnally embark on the task of proving the excision axiom for singular homology. 24 Chapter 3. Singular Homology

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 a topological space and U ⊂ PX is a collection of subsets of X, whose interiors n o ˚ ˚ U := U U ∈ U cover X. An arrow f :(X, U) → (Y, V) consists of a map f : X → Y such −1 −1 that U = f V := f V V ∈ V . We equip this category with the models n o q ˚ M := (∆ , U) q ∈ N, U ⊂ PX arbitrary, such that U covers X . We define two functors S0,S00 : Cov → Ch(AbGrp) on the objects by S0(X, U) := SX, 00 0 00 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 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. S0 and S00 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 p ˚ p Mp := (∆ , U) U ⊂ PX, such that U covers ∆ ⊂ M

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

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

0 is a basis for SpX and so S is free with models M. By the same argument (with ep := 1SU 00 instead of 1S∆p ) S is free with models M. Clearly S0 is aacyclic on M and so is S00 by the lemma (4.4) above, which proves our proposition.

00 ∼ 0 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 by corollary (2.3) in chapter2 this can be lifted to a natural isomorphism ϕ˜ : S00 =∼ S0 and as an immediate consequence we get the following corollary.

(5.3) Corollary. For X a topological space and U ⊂ PX such that U˚ 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 ⊂ 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. Paragraph 5. Excision 25

Proof. Consider U := {A, X \U} which satisﬁes the condition that U˚ 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

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. 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 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, New York, 1994 INDEX

Numbers Cone, 19 Connecting morphism,5 1,4

D A Dimension Aacyclic chain complex, 14 invariance of, 12 Aacyclic functor on models, 14 Absolute homology,5 E Absolute singular homology, 18 Acyclic Model Theorem, 14 Excision,6 Admissible for singular homology, 24 category for homology theory,4 Affine singular simplex, 19 Almost acyclic functor F on models, 14 Face of a singular simplex, 17 B Free functor with models, 13 Barycentre, 20 Barycentric subdivision, 20 Boundary, 17 H Boundary morphism, 17 Boundary operator,5 Homology Braid diagram,8 of a pair,5 Braid Lemma,8 of a space,5 of spheres, 12 C reduced,9 Homology sequence,6 Category of a triple,9 admissible for homology theory,4 reduced, 10 Chain, 17 Homology theory,5 affine, 21 Homotopic U-small, 23 maps in Top(2),5 Chain complex natural transformations, 14 aacyclic, 14 Homotopy Chain homotopic in Top(2),5 natural transformations, 14 Homotopy Invariance,5 Coefficients Homotopy invariance of a homology theory,6 of singular homology, 18 28 Index

I relative, 18 Singular p-chain, 17 I,4 Singular simplex, 17 Inclusion,3 Space,3 Invariance Standard models, 19 of dimension, 12 Standard simplex, 17 Subdivision L barycentric, 20

Lattice T of a pair,4 Long Exact Homology Sequence,6 Top,3 of a triple,9 Top(2),3 reduced, 10 Top(3),3

M U

Map,3 U-small chain, 23 Models U-small simplex, 23 of a category, 13

V N v-cone, 19 Naturally chain homotopic, 14 Naturally homotopic, 14

Ordinary homology theory,6

Reduced homology,9 Relative homology,5 of discs, 12 Relative singular homology, 18 Restriction functor,5

Shift functor, 14 Simplex, 17 aﬃne singular, 19 standard, 17 U-small, 23 Singular homology, 18 absolute, 18