07-17-2015 Contents 1. Summary of the Homology of Posets 1 2

07-17-2015

Contents 1. Summary of the Homology of Posets 1 2. Deﬁnition of Singular Homology 3 3. The Hurewicz homomorphism 6 4. Some Properties of Singular Homology 7

1. Summary of the Homology of Posets By now, you have realized that what algebraic topologists do is assign invariants to topological spaces which do not depend on the homotopy type. That is, we assign quantities F (X) so that if X ' Y , then F (X) =∼ F (Y ). Further, we have the language of functors to make this more precise. What we’ve done so far is define functors F : Top → C where C is some algebraic category (abelian groups, chain complexes, . . . ), such that if X ' Y then F (X) ' F (Y ) (in the case of chain complexes) or better F (X) =∼ F (Y ) (in the case of homology). Definition 1.1. Let T be some subcategory of topological spaces. A homotopy functor is a functor F : D → C such that if f, g : X → Y are homotopic, i.e., f ' g, then F (f) = F (g). Exercise 1.2. If F is a homotopy functor and X ' Y , then F (X) =∼ F (Y ). Our first examples were

π1 : Top∗ → Groups, and for n ≥ 2, πn : Top∗ → AbGroups . Now, we deﬁned the homology of a poset, which gave functors P Hn : A-spaces → Ab .

If we think in terms the category Ab∗ of sequences of abelian groups (deﬁne the maps!), we can write this as P H∗ : A-spaces → Ab∗ . We also proved that Proposition 1.3. Let f, g : X → Y be maps of A-spaces. If f ≤ g, then f and g induce the same map on homology, that is P P P P H∗ (f) = H∗ (g): H∗ (X) → H∗ (Y ). However, we only have the following partial result: 1 2 07-17-2015

Theorem 1.4. If X and Y are A-spaces and f, g : X → Y satisfy f ≤ g, then f ' g. If X and Y are F -spaces, we have a converse. That is, in this case, if f ' g, then there are maps f1, . . . , fn such that,

f ≥ f1 ≤ f2 ≥ f3 ≤ f4 ≥ ... ≤ fn ≥ g. P Therefore, we have not shown that H∗ is a homotopy functor. However, we have shown that its restriction to F -spaces is such:

P Proposition 1.5. H∗ : F -spaces → Ab∗ is a homotopy functor. Deﬁnition 1.6. We also deﬁned, for A ⊂ X, the relative homology of X with respect to A d ker Cn(X)/Cn(A) −→ Cn−1(X)/Cn−1(A) HP (X,A) = n d im Cn+1(X)/Cn+1(A) −→ Cn(X)/Cn(A)

= n’th homology of the chain complex C∗(X)/C∗(A)

Therefore, at least on F -spaces, we’ve proved that H∗ has the following properties. (Property (3) is an exercise)

P Proposition 1.7. Let H∗ : F -spaces → Ab be the poset homology functor. Then P P (1) (Homotopy Invariance) If f ' g, then H∗ (f) = H∗ (g) P P (2) (Dimension Axiom) H0 (∗) = Z and Hn (∗) = 0 if n > 0. ` (3) (Additivity) If X = Xi, where Xi are the diﬀerent path components of P ∼ L P X, then H∗ (X) = i H∗ (Xi). (4) (Exactness) If A ⊂ X, there is a long exact sequence P P P P ... → Hn (A) → Hn (X) → Hn (X,A) → Hn−1(A) → ... We also proved that: (a) (Mayer-Vietoris) If A and B are sieves in X and X = A ∪ B, there is a long exact sequence P P P P P ... → Hn (A ∩ B) → Hn (A) ⊕ Hn (B) → Hn (X) → Hn−1(A ∩ B) → ... (b) (Suspension isomorphism) There is an isomorphism

( P P ∼ H1 (ΣX) ⊕ Z n = 0 Hn (X) = P Hn+1(ΣX) n > 0 (Note that except for (1), these all hold for A-spaces.) Remark 1.8. A little later, we will give an axiomatic description for functors which satisfy the above properties. (1)-(4) will be axioms for such functors. However, the list of axioms is still incomplete and (4) is rather useless since it is not clear now P that H∗ (X,A) has (1) any geometric meaning that makes it interesting to compute or P P (2) is easier to compute than H∗ (X) and/or H∗ (A) so that it could be used as a tool in understanding these terms. P P We need a property that allows us to identify H∗ (X,A) with H∗ (X/A) in favorable cases. We ask ourselves: 07-17-2015 3

Question 1.9. Is there a functor H∗ : Top → Ab which satisﬁes all the above P ∼ properties and such that H∗ (X) = H∗(X)?

2. Definition of Singular Homology To deﬁne the homology of a space, we must associate to a space X a chain complex C∗(X). Then H∗(X) will simply be its homology. Recall that the n- simplex ∆n is the subspace of Rn+1 given by n n+1 ∆ = {(t0, . . . , tn) ∈ R | 0 ≤ ti ≤ 1, t0 + ... + tn = 1} Now, note that there are n + 1 copies of ∆n−1 in ∆n. These are the faces of ∆n. We denote them by n n ∆i = {(t0, . . . , ti−1, 0, ti+1, . . . , tn) ∈ ∆ }

Remark 2.1. Let xi = (t0 = 0, . . . , ti−1 = 0, ti = 1, ti+1 = 0, . . . , tn = 0). Then n n x0, . . . , xn are the vertices of ∆ . The face ∆i is the n-simplex with vertices x0,..., xbi, . . . , xn. There’s a canonical homeomorphism: n−1 n ∆ → ∆i which sends (t0, . . . , tn−1) to (t0, . . . , ti−1, 0, ti, . . . , tn)

Deﬁnition 2.2. A singular n-simplex on X is a continuous map σ : ∆n → X. Let Xn be the set of singular n-simplicies:

Xn = {σ : ∆n → X}

Given a singular n-simplex on X, σ : ∆n → X, we can obtain n + 1 singular n − 1-simplices by restricting σ to the faces of ∆n. That is, n σ| n : ∆ ∼ ∆ → X ∆i n−1 = i

Deﬁnition 2.3. The group of singular n-chains on X, denoted Cn(X), is the free abelian group generated by the set of all singular n-simplices, that is ( k ) X Cn(X) = Z{Xn} = aiσi | σi ∈ Xn, ai ∈ Z i=0

Further, let ∂ : Cn(X) → Cn−1(X) be deﬁned on a generator σ ∈ Cn(X) by n X i ∂ (σ) = (−1) σ| n n ∆i i=0

One veriﬁes that ∂ ◦ ∂ = 0. The chain complex C∗(X) is called the singular chains on X. Its homology is called the singlular homology of X, denoted

ker ∂n Hn(X) = im ∂n+1

If f : X → Y , we get a map C∗(f): C∗(X) → C∗(Y ). For a singular n-simplex σ : ∆n → X, we let Cn(f)(σ) = f ◦ σ ∈ Cn(Y ). This induces a map H∗(f): H∗(X) → H∗(Y ). 4 07-17-2015

Hence, singular homology is a functor:

H∗ : Top → Ab∗ Remark 2.4. Note that

H0(X) = C0(X)/(im(∂1 : C1(X) → C0(X))). A singular 0-simplex is a function σ : ∆0 = {0} → X, so it is just a point in X. We let 0 σx : ∆ → X be the simplex with σx(0) = x. Therefore, ∼ C0(X) = Z{X} where Z{X} is the free abelian group on X. An isomorphism is given by k k k X X X aiσxi 7→ aiσxi (0) = aixi. i=0 i=0 i=0

So, we can think of the elements of C0(X) as formal sums k X aixi i=0 where the points xi ∈ X and ai ∈ Z. Further, suppose that two points x0, x1 ∈ X are in the same path component, that is, that there is a path σ : I → X with σ(0) = x0 and σ(1) = x1. Since 1 I = ∆ , we have σ ∈ C1(X). Further,

∂ (σ) = σ| 1 − σ| 1 1 ∆0 ∆1

= σx1 − σx0

7→ x1 − x0.

Therefore, in C0(X)/ im(∂1), we have the relation that x1 = x0 for any two generators which are connected by a path. Therefore, any two generators for C0(X) which are in the same path components are identiﬁed when we pass to the quotient H0(X) = C0(X)/ im(∂1). Further, these are the only relations, so H0(X) is a free abelian group with one generator for each path component of X. This proves: ` Proposition 2.5. If X = α∈I Xi where Xi are the distinct path components of X, then ∼ M H0(X) = Z. α∈I An explicit isomorphism is induced by the map M C0(X) → Z α∈I given by k ! X X aixi 7→ ai i=0 xi∈Xα α∈I ∼ Proposition 2.6 (Dimension Axiom). H0(∗) = Z and Hn(∗) = 0 if n > 0. 07-17-2015 5

n Proof. We have Cn(∗) = Z since there is only one map σn : ∆ → ∗, namely, the constant map at ∗. Further, since

σ | n = σ n ∆i n−1 we have for n > 0

n X i ∂ (σ) = (−1) σ| n n ∆i i=0 n X i = (−1) σn−1 i=0 ( 0 n is odd = σn−1 n is even

The chain complex thus looks like

0 ∼= 0 ∼= 0 ∼= 0 ... → Z −→ Z −→ Z −→ Z −→ Z → ... → Z −→ Z −→ Z −→ Z → 0

Remark 2.7. We can deﬁne a related functor

He∗ : Top∗ → Ab called the reduced homology as follows. For a based space X, there are maps

p ∗ −→i X −→∗

These induce maps on homology

i p H∗(∗) / H∗(X) / H∗(∗)

Z / H∗(X) / Z

This implies that H0(X) has a direct summand isomorphic to Z. We deﬁned He∗(X) to be the other summand, ∼ H∗(X) = Z ⊕ He∗(X). Therefore, ∼ He∗(X) = H∗(X)/H∗(∗). If X is path connected, the reduced homology is the same as the homology of X in positive degrees. All we did is remove a copy of Z in degree zero: ( He0(X) ⊕ Z n = 0 Hn(X) = Hen(X) n > 0. 6 07-17-2015

3. The Hurewicz homomorphism Let X be a path connected based space with base point ∗. Note that a based 1 1 map f : S → X can be viewed as a map σf : ∆ → X where σ(0) = σ(1) = ∗. Further, since σf (0) = σf (1),

∂1(σf ) = σf (1) − σf (0) = σ∗ − σ∗ = 0.

Therefore, σf ∈ C1(X) is a cycle, so it represents an element in H1(X). Further, after today, you should be able to do the following exercise:

1 Exercise 3.1. Prove that if f ' g : S → X are homotopic, then [σf ] = [σg] ∈ 1 H1(X). (Hint: consider the maps H∗(f),H∗(g): H∗(S ) → H∗(X).) Once we’ve done that exercise, we get a map

h : π1(X) → H1(X) This map is called the Hurewicz homomorphism. We won’t prove it right now, but ∼ π1(X)/[π1(X), π1(X)] = H1(X), that is H1(X) is the abelianization of π1(X). Remark 3.2. For any group G,[G, G] is the normal subgroup of G generated by the elements of the form ghg−1h−1, for g, h ∈ G. Note that, in G/[G, G], the residue class of ghg−1h−1 is equal to that of the identity e. This implies that gh and hg are equivalent modulo [G, G] for all g, h ∈ G. Therefore, G/[G, G] is abelian. Example 3.3.

1 1 • We’ve seen that π1(S ∨ S ) = Z ∗ Z the free group on two generators. This 1 1 implies that H1(S ∨S ) = Z⊕Z, the free abelian group on two generators. • If π1(X) is already abelian, then h : π1(X) → H1(X) is an isomorphism. In fact, we can do this for higher homotopy groups also: Deﬁnition 3.4 (Hurewicz homomorphism). Let X be a based topological space. n n Let f : S → X be a continuous map and σf : ∆ → X be the singular n-simplex obtained from f by identifying Sn =∼ ∆n/∂(∆n). There is a map h : πn(X) → Hn(X) deﬁned by h([f]) = [σf ] which is a group homomorphism for all n such that n ≥ 1.

Theorem 3.5 (Hurewicz). Let X be a based space. Suppose that πk(X) = 0 when 0 ≤ k ≤ n − 1. Then h : πn(X) → Hn(X) is an isomorphism if n > 1 and the abelianization if n = 1. Remember that homotopy groups are hard to compute, but homology is not. This theorem is important since it identiﬁes something hard, πn(X), with something easy to compute, Hn(X). 07-17-2015 7

4. Some Properties of Singular Homology Definition 4.1. A pair (X,A) of topological spaces is a space X with a subspace A ⊂ X. A map f :(X,A) → (Y,B) of pairs is a map f : X → Y such that f(A) ⊂ B. Such a map is a (weak) homotopy equivalence if it induces (weak) homotopy equivalences X → Y and A → B. Maps of pairs f, g :(X,A) → (Y,B) are homotopic if there is a map of pairs F :(X × I,A × I) → (Y,B) with F (0) = f and F (1) = g. As for poset homology, we can define relative homology groups for singular homology: Definition 4.2. Let (X,A) be a pair. The relative singular homology of X with respect to A, denoted Hn(X,A), is defined as follows. Let

Cn(X,A) = Cn(X)/Cn(A) with diﬀerential induced by that on C∗(X). Then

ker(∂ : Cn(X,A) → Cn−1(X,A)) Hn(X,A) = . im(∂ : Cn+1(X,A) → Cn(X,A)) Remark 4.3. It will be convenient to study singular homology as a functor

H∗ : pairs in Top → Ab∗ We note that H∗(X) = H∗(X, ∅) since C∗(∅) = 0 so that

C∗(X, ∅) = C∗(X)/C∗(∅) = C∗(X). Therefore, we lose nothing by considering pairs. Proposition 4.4 (Exactness). There is a long exact sequence

... → Hn(A) → Hn(X) → Hn(X,A) → Hn−1(A) → ... This exact sequence is natural. Given a map of pairs : (X,A) → (Y,B), we get an induced map of long exact sequences. Proof. By deﬁnition, there is an exact sequence of chain complexes:

0 → C∗(A) → C∗(X) → C∗(X,A) → 0 Then we apply the snake lemma. Exercise 4.5. If X is a based space, then ∼ He∗(X) = H∗(X, ∗) ` Proposition 4.6 (Additivity). If X = (Xi,Ai), then ∼ M H∗(X,A) = H∗(Xi,Ai). i Proof. Exercise.

Deﬁnition 4.7. A homotopy between maps of chain complexes f∗, g∗ : A∗ → B∗ is a sequence of homomorphisms Pn : An → Bn+1 such that

fn − gn = ∂Pn + Pn−1∂ 8 07-17-2015

Theorem 4.8. If f ' g :(X,A) → (Y,B), then H∗(f) = H∗(g). Proof. We sketch the argument when A, B = ∅. Given a homotopy F : X × I → Y from f to g, we use it to construct a homotopy Pn : Cn(X) → Cn+1(Y ) where

Cn(f) − Cn(g) = ∂Pn + Pn−1∂ in a way very similar to that Inna used to construct P when she was proving that if P P n f ≤ g, then H∗ (f) = H∗ (g). The key point is to split ∆ × I into n + 1-simplices and this is done exactly like Inna’s proof for posets. This argument can also be made to work for other choices of A and B.