University of Cambridge

Mathematics Tripos

Part II Algebraic Topology

Michaelmas, 2018

Lectures by H. Wilton

Notes by Qiangru Kuang Contents

Contents

0 Introduction 2

1 The fundamental group 3 1.1 Deforming maps and spaces ...... 3 1.2 The fundamental group ...... 4

2 Covering spaces 8 2.1 Definition and first examples ...... 8 2.2 Lifting properties ...... 9 2.3 Applications to calculations of fundamental groups ...... 11 2.4 The fundamental group of 푆1 ...... 12 2.5 Existence of universal covers ...... 13 2.6 The Galois correspondence ...... 14

3 Seifert-van Kampen theorem 16 3.1 Free groups and presentations ...... 16 3.2 Seifert-van Kampen theorem for wedges ...... 18 3.3 Seifert-van Kampen theorem ...... 19 3.4 Attaching cells ...... 20 3.5 Classification of surfaces ...... 22

4 Simplicial complexes 24 4.1 Simplices and stuff ...... 24 4.2 Barycentric subdivision ...... 25 4.3 Simplicial approximation theorem ...... 27

5 Homology 29 5.1 Simplicial homology ...... 29 5.2 Chain complexes & chain homotopies ...... 32 5.3 Homology of the simplex and the sphere ...... 35 5.4 Continuous maps and homotopies ...... 35

6 Homology calculations 38 6.1 Homology of spheres and applications ...... 38 6.2 Mayer-Vietoris theorem ...... 38 6.3 Homology of compact surfaces ...... 42 6.4 Rational homology and Euler characteristic ...... 44 6.5 Lefschetz fixed-point theorem ...... 45

Index 47

1 0 Introduction

0 Introduction

Question. Is the Hopf link really linked? More formally, is there a homeomor- phism R3 → R3 taking 퐻 to 푈?

퐻 can be realised as 푆1 ⨿ 푆1 → R3. For 푈, we can consider 푆1 ⨿ 푆1 as boundary of 퐷1 ⨿ 퐷1 and the map extends to a map to discs. So it makes sense to phrase the question as

Question. Does the Hopf link 휂 ∶ 푆1 ⨿ 푆1 → R3extend to a map of discs? This is an example of an extension problem. 푛 푆푛−1 ∶= {푥 ∈ 푛 ∶ ∑푛 푥2 = Here is another example. Define the -sphere R 푖=1 푖 1} 퐷푛 = {푥 ∈ 푛 ∶ ∑푛 푥2 = 1} , which sits inside R 푖=1 푖 . We can ask: 푛−1 푛−1 푛 Question. Does the identity map id푆푛−1 ∶ 푆 → 푆 factor through 퐷 ? To gain some intuition, let’s consider small 푛. For 푛 = 1, 푆0 = {−1, 1}. The answer is no by Intermediate Value Theorem, or connectedness from topology. For 푛 = 2, this answer is again no by winding number argument. What about 푛 ≥ 3? These problems are hard because we have to consider continuous maps be- tween two spaces, which are in general very big and hard to compute. On the other hand, a comparable algebraic problem is

Question. Does the map id ∶ Z → Z factor through 0? Well that’s much much easier!

2 1 The fundamental group

1 The fundamental group

Throughout this course, “maps” mean continuous maps.

1.1 Deforming maps and spaces

Definition (homotopy). Let 푓0, 푓1 ∶ 푋 → 푌 be maps. A homotopy between 푓0 and 푓1 is a map 퐹 ∶ 푋 × [0, 1] → 푋 such that 퐹 (푥, 0) = 푓0(푥) and 퐹 (푥, 1) = 푓1(푥) for all 푥 ∈ 푋. If 퐹 exists, we say that 푓0 is homotopic to 푓1 and write 푓0 ≃ 푓1, or to emphasise the homotopy, 푓0 ≃퐹 푓1.

Notation. 퐼 = [0, 1]. We often write 푓푡(푥) = 퐹 (푥, 푡). 푛 Example. If 푌 is a convex region in R then for any 푓0, 푓1 ∶ 푋 → 푌, the straightline homotopy 퐹 (푥, 푦) = 푡푓1(푥) + (1 − 푡)푓0(푥) is a homotopy 푓0 ≃ 푓1.

Definition (relative homotopy). If 푍 ⊆ 푋 and 퐹 (푧, 푡) = 푓0(푧) = 푓1(푧) for all 푧 ∈ 푍, 푡 ∈ 퐼, then 퐹 is a homotopy relative to 푍, write 푓0 ≃퐹 푓1 rel 푍.

Lemma 1.1. The relation ≃ (rel 푍) is an equivalence relation on maps 푋 → 푌. 푓 ≃ Proof. Reflexivity and symmetry are easy. For transitivity, suppose 0 퐹0 푓 ≃ 푓 1 퐹1 2. Let 1 퐹0(푥, 2푡) 푡 ≤ 2 퐹 (푥, 푡) = { 1 퐹1(푥, 2푡 − 1) 푡 ≥ 2 which is the homotopy we need.

Definition (homotopy equivalence). 푓 ∶ 푋 → 푌 and 푔 ∶ 푌 → 푋 is a homotopy equivalence if 푔 ∘ 푓 ≃ id푋 and 푓 ∘ 푔 ≃ id푌. In this case we say 푋 is homotopy equivalent to 푌 and write 푋 ≃ 푌.

Example. Let 푋 = ∗, the space with one point and 푌 = R푛. Let 푓 ∶ ∗ ↦ 0, 푔 be the unique map 푌 → 푋. Then 푔 ∘ 푓 = id푋, and 푓 ∘ 푔 = 0 ≃ id푌 via the straightline homotopy. Therefore R푛 is homotopy equivalent to ∗.

Definition (contractible). A space 푋 is contractible if 푋 ≃ ∗.

Example. Let 푋 = 푆1, 푌 = R2 − {0}. Let 푓 ∶ 푋 → 푌 be the natural inclusion 푥 nad 푔 ∶ 푌 → 푋, 푥 ↦ ‖푥‖ . Then

푔 ∘ 푓 = id푋 푥 푓 ∘ 푔(푥) = ∈ 2 ‖푥‖ R Although 푌 is not convex, for all 푥, 푡, straightline homotopy 퐹 (푥, 푡) between 푓 ∘푔 and id푌 satisfies 퐹 (푥, 푡) ≠ 0 so 푓 ∘ 푔 ≃퐹 id푌. Thus 푋 ≃ 푌.

3 1 The fundamental group

Definition (retract, deformation retract). Let 푓 ∶ 푋 → 푌 and 푔 ∶ 푌 → 푋. If 푔 ∘ 푓 = id푋 then 푋 is a retract of 푌. If in addition 푓 ∘ 푔 ≃ id푌 rel 푓(푋) then we say 푋 is a deformation retract of 푌.

Note that whenever we have 푔 ∘ 푓 = id푋, 푓 is injective so we can think 푋 as being embedded in 푌. Informally, 푌 is “as complicated” as 푋.

Lemma 1.2. Homotopy equivalence is an equivalence on topological spaces.

Proof. Symmetry and reflexivity are obvious. For transitivity, consider

푓 푓 푋 푌 푍 푔 푔

′ ′ Need to show that 푔 ∘ (푔 ∘ 푓 ) ∘ 푓 ≃ id푋 (and the other direction will follow ′ ′ similarly). By hypothesis 푔 ∘ 푓 ≃퐹 ′ id푌. Now

푔(퐹 ′(푓(푥), 푡)) is a homotopy ′ ′ 푔 ∘ 푔 ∘ 푓 ∘ 푓 ≃ 푔 ∘ id푌 ∘푓 = 푔 ∘ 푓 ≃ id푋 .

1.2 The fundamental group

Definition (path, loop). A path (from 푥0 to 푥1) is a continuous map 훾 ∶ 퐼 → 푋 (with 훾(0) = 푥0, 훾(1) = 푥1). A loop (based at 푥0) is a path from 푥0 to 푥0.

Definition (homotopy of path). Let 훾0, 훾1 be paths from 푥0 to 푥1.A homotopy (of path) from 훾0 to 훾1 is a homotopy

훾0 ≃퐹 훾1 rel{0, 1}.

Definition (concatenation of path, constant path, inverse path). Let 훾 be a path from 푥 to 푦 and 훿 a path from 푦 to 푧. 1. The concatenation of 훾 and 훿 is

1 훾(2푡) 푡 ≤ 2 (훾 ⋅ 훿)(푡) = { 1 훿(2푡 − 1) 푡 ≥ 2

2. The constant path (at 푥) is 푐푥(푡) = 푥. 3. The inverse path to 훾 is 훾(푡) = 훾(1 − 푡).

4 1 The fundamental group

Theorem 1.3 (fundamental group). Let 푥0 ∈ 푋. Let

휋1(푋, 푥0) = {loops based at 푥0}/ ≃ .

This has a group structure with • [훾][훿] = [훾 ⋅ 훿], [푐 ] • identity 푥0 , • [훾]−1 = [훾].

We call 휋1(푋, 푥0) the fundamental group of 푋 (based at 푥0).

Proof. To prove the theorem, we need to check that multiplication and inverses are well-defined and the group axioms are satisfied.

Lemma 1.4. If 훾0, 훾1 are paths to 푦 and 훿0, 훿1 are paths from 푦 and 훾0 ≃ 훾1, 훿0 ≃ 훿1, then 훾0 ⋅ 훿0 ≃ 훾1 ⋅ 훿1. 훾 ≃ 훾 Also 0 1.

Proof. We only show for concatenation. Inverses are similar. Let 훾0 ≃퐹 훾1, 훿0 ≃퐺 훿1. (proof by picture) Algebraically, the homotopy is given by

1 퐹 (푠, 2푡) 푡 ≤ 2 퐻(푠, 푡) = { 1 퐺(푠, 2푡 − 1) 푡 ≥ 2

Now we check that the group axioms are satisfied.

Lemma 1.5. 1. (훼 ⋅ 훽) ⋅ 훾 ≃ 훼 ⋅ (훽 ⋅ 훾).

2. 훼 ⋅ 푐푥 ≃ 훼 ≃ 푐푤 ⋅ 훼.

3. 훼 ⋅ 훼 ≃ 푐푤.

Proof. We show 1. The other two are similar. Let

⎧훼(3푡) 푡 ≤ 1 { 3 훿 = 훽(3푡 − 1) 1 ≤ 푡 ≤ 2 ⎨ 3 3 { 2 ⎩훾(3푡 − 2) 3 ≤ 푡 ≤ 1 Let 4 1 3 푡 푡 ≤ 2 푓0(푡) = { 1 2 1 3 + 3 푡 푡 ≥ 2 and 2 1 3 푡 푡 ≤ 2 푓1(푡) = { 1 4 1 − 3 + 3 푡 푡 ≥ 2

5 1 The fundamental group

Note that 푓0 ≃ 푓 as paths via the straightline homotopy in 퐼. But

(훼 ⋅ 훽) ⋅ 훾 = 훿 ∘ 푓0

훼 ⋅ (훽 ⋅ 훾) = 훿 ∘ 푓1 so they are homotopic as path.

푛 푛 Example. Let 푋 = R , 푥0 = 0. Consider a loop 훾 in R based at 0. The 푛 straightline homotopy shows that 훾 ≃ 푐0 as path. Therefore 휋1(R , 0) ≅ 1.

Lemma 1.6. Let 푓 ∶ 푋 → 푌 be such that 푓(푥0) = 푦0. There is a well-defined homomorphism

푓∗ ∶ 휋1(푋, 푥0) → 휋1(푌 , 푦0) [훾] ↦ [푓 ∘ 훾]

Furthermore,

′ ′ 1. if 푓 ≃ 푓 rel{푥0} then 푓∗ = 푓∗.

2. if 푔 ∶ 푌 → 푍 is another map then 푓∗ ∘ 푔∗ = (푓 ∘ 푔)∗. ( ) = 3. id푋 ∗ id휋1(푋,푥0).

Proof. Easy.

We’d like to eliminate the dependence of 휋1(푋, 푥0) on 푥0, at least when 푋 is path-connected. Suppose 푥0, 푥1 ∈ 푋. What do 휋1(푋, 푥0) and 휋1(푋, 푥1) have to do with each other, where 푋 is path-connected? Fix 훼 a path from 푥0 to 푥1.

Lemma 1.7. There is a well-defined group homomorphism

훼# ∶ 휋1(푋, 푥0) → 휋1(푋, 푥1) [훾] ↦ [훼 ⋅ 훾 ⋅ 훼]

Furthermore

′ ′ 1. if 훼 ≃ 훼 then 훼# = 훼#, (푐 ) = 2. 푥0 # id휋1(푋,푥0),

3. if 훽 is a path from 푥1 to 푥2, 훽# ∘ 훼# = (훼 ⋅ 훽)#.

4. if 푓 ∶ 푋 → 푌 then (푓 ∘ 훼)# ∘ 푓∗ = 푓∗ ∘ 훼#.

Now it makes sense to talk about isomorphism type of the fundamental group of a path-connected space.

6 1 The fundamental group

Definition (simply connected). If 푋 is path-connected and 휋1(푋, 푥0) ≅ 1 for some (i.e. any) 푥0 ∈ 푋 then we say 푋 is simply connected.

Our last task is to understand what homotopies that don’t fix basepoints do to the fundamental group.

Lemma 1.8. Suppose 푓, 푔 ∶ 푋 → 푌 is such that 푓 ≃퐹 푔. Define 훼(푡) = 퐹 (푥0, 푡), a path from 푓(푥0) to 푔(푥0). Then the following diagram commutes:

휋1(푌 , 푓(푥0))

푓∗

훼 휋1(푋, 푥0) #

푔∗

휋1(푌 , 푔(푥0))

i.e. 푔∗ = 훼# ∘ 푓∗.

Proof. Let [훾] ∈ 휋1(푋, 푥0). We need to show that

[푔 ∘ 훾] = 푔∗[훾] = 훼# ∘ 푓∗[훾] = [훼 ⋅ (푓 ∘ 훾) ⋅ 훼] which is saying 푔 ∘ 훾 ≃ 훼 ⋅ (푓 ∘ 훾) ⋅ 훼 as paths. Consider

퐼 × 퐼 → 푌 (푠, 푡) ↦ 퐹 (훾(푠), 푡)

Let 퐻 be the straightline homotopy in 퐼 × 퐼 between the yellow path and the brown path. Then 퐺 ∘ 퐻 is the homotopy we need.

Theorem 1.9. If 푓∶푋→푌,푔∶푌 →푋 is a pair of homotopy equivalences and 푥0 ∈ 푋 then 푓∗ ∶ 휋1(푋, 푥0) → 휋1(푌 , 푓(푥0)) is an isomorphism.

Proof. Suffices to prove that 푓∗ is bijective. Let 푔 ∘ 푓 ≃퐹 id푋 and 훼 be the path defined from 퐹 as above. Then

푔 ∘ 푓 = (푔 ∘ 푓) = 훼 ∘ = 훼 ∗ ∗ ∗ # id휋1(푋,푥0) # so 푓∗ is injective. Similarly it is surjective.

Corollary 1.10. Contractible spaces are simply connected.

7 2 Covering spaces

2 Covering spaces

2.1 Definition and first examples

Definition (covering space). Let 푝 ∶ 푋̂ → 푋 be a map. An open set 푈 ⊆ 푋 is evenly covered if there is a discrete space Δ푈 and an identification −1 −1 푝 (푈) = Δ푈 × 푈 such that on 푝 (푈), 푝 coincides with projection to the second factor. If every 푥 ∈ 푋 has an everly covered neighbourhood, we say that 푝 is a covering map and 푋̂ is a covering space

−1 Alternatively, write 푈훿 = {훿} × 푈. Then 푝 (푈) = ∐ 푈훿. Write 푝|훿 = 훿∈Δ푈 푝| 푈훿 which is a homeomorphism. Example. 1. Let 푋̂ = R, 푋 = 푆1 and define 푝 ∶ R → 푆1 푡 ↦ 푒2휋푖푡 Let 1 ∈ 푈 ⊊ 푆1. Choose a branch of log well-defined on 푈 such that log 1 = 0. Every point ̂푧∈ 푝−1(푈) can be written uniquely as log(푧) ̂푧= 푘 + 2휋푖 where 푧 = 푝( ̂푧) ∈ 푈 and 푘 ∈ Z, i.e. 푝−1(푈) = Z × 푈. Thus 푈 is evenly covered. The same proof shows that 푝 is a covering map. 2. Let 푋̂ = 푋 = 푆1. Define 1 1 푝푛 ∶ 푆 → 푆 푧 ↦ 푧푛 This is also a covering map by essentially the same proof by choosing a 푛th root of unity. In this case Δ푛 is the 푛th roots of unity.

3. Let 푋̂ = 푆2 and 퐺 = Z/2Z acts on 푆2 via the antipodal map. Let 푋 = 푋/퐺̂ = {{푥, −푥} ∶ 푥 ∈ 푆2}

and 푝 ∶ 푋̂ → 푋 be the quotient map. The orbit space 푋 can be identified with straightlines in R3 passing through the origin. Given a line ℓ through the origin, let 2 퐶ℓ = {푦 ∈ 푆 ∶ 푦 perpendicular to ℓ}. 2 Then 푆 − 퐶ℓ = 푈+ ⨿ 푈−. Let 푈 = 푝(푈+ ⨿ 푈−), an open neighbourhood ℓ 푋 푝| 푝| 푈 of in . Note that 푈+ and 푈− are both homeomorphisms onto . Thus 푈 is evenly covered and 푝 is a covering map. 푋 = R푃 2 is the real projective plane. Note that in all three examples, for all points 푥 ∈ 푋, the number of copies of 푈 in 푝−1(푈) is the same. We give a name to such covering spaces:

8 2 Covering spaces

Definition (푛-sheeted). A covering map 푝 ∶ 푋̂ → 푋 is 푛-sheeted where 푛 ∈ N ∪ {∞} if for all 푥 ∈ 푋, #푝−1(푥) = 푛.

2.2 Lifting properties

Let 푝 ∶ 푋̂ → 푋 be a covering map throughout the section.

Definition (lift). A lift of 푓 ∶ 푌 → 푋 to 푋̂ is a map 푓̂ ∶ 푌 → 푋̂ such that 푓 = 푝 ∘ 푓,̂ i.e. the following diagram commutes:

푋̂ 푓 ̂ 푝 푓 푋 푋

Lemma 2.1 (uniqueness of lift). Suppose 푓 ∶ 푌 → 푋 where 푌 is connected ̂ ̂ ̂ and locally path-connected. Let 푓1, 푓2 ∶ 푌 → 푋 are both lifts of 푓. If there ̂ ̂ ̂ ̂ exists 푦 ∈ 푌 such that 푓1(푦) = 푓2(푦) then 푓1 = 푓2.

Proof. Consider ̂ ̂ 푆 = {푦 ∈ 푌 ∶ 푓1(푦) = 푓2(푦)}. Claim that 푆 is both open and closed, from which the lemma follows imme- diately. Given 푦0 ∈ 푌, let 푈 be an evenly covered neighbourhood of 푓(푦0) −1̂ and 푉 ⊆ 푓 (푈) a path-connected neighbourhood of 푦0. Let 푦 ∈ 푉 be ar- bitrary. Need to show that 푦0 ∈ 푆 if and only if 푦 ∈ 푆. If 푦0 ∈ 푆 then ̂ ̂ 푓1(푦0) = 푓2(푦0) ∈ 푈훿 for some 훿 ∈ Δ푈. Let 훼 be a path in 푉 from 푦0 to 푦. Then ̂ −1 ̂ 푓 ∘ 훼 is a path from 푓(푦0) to 푓(푦). Then 푓푖 ∘ 훼 is a path in 푝 (푈) from 푓푖(푦0) ̂ ̂ ̂ ̂ to 푓푖(푦). It follows that 푓푖(푦) ∈ 푈훿 so 푓1(푦) = (훿, 푓(푦)) = 푓2(푦) so 푦 ∈ 푆. The converse is identical.

Definition (lift at a point). Let 훾 ∶ 퐼 → 푋 be a path with 훾(0) = 푥0.A ̂ −1 (unique) lift of 훾 to 푋 such that ̂훾(0) =0 ̂푥 ∈ 푝 (푥0) is called the lift of 훾 at 0̂푥 .

Lemma 2.2 (path-lifting lemma). Let 훾 ∶ 퐼 → 푋 be a path with 훾(0) = 푥0. −1 For any 0̂푥 ∈ 푝 (푥0) there is a uniqueness ̂훾 of 훾 at 0̂푥 .

Proof. Uniqueness follows from the more general uniquenss of lift so suffices to show existence. Consider

푆 = {푡 ∈ 퐼 ∶ lift of 훾|[0,푡] at 0̂푥 exists}, as 0 ∈ 푆, the lemma follows if we can show 푆 is both open and closed. Let 푡0 ∈ 퐼. Then 훾(푡0) ∈ 푈 for some evenly covered neighbourhood 푈. There exists a path- connected neighbourhood 푉 of 푡0 such that 훾(푉 ) ⊆ 푈. Let 푡 ∈ 푉. We’ll prove that 푡0 ∈ 푆 if and only if 푡 ∈ 푆. By symmetry suffices to show one direction.

9 2 Covering spaces

Suppose 푡0 ∈ 푆, 푡 ∉ 푆. Since 푡0 ∈ 푆, ̂훾(푡0) is well-defined so let ̂훾(푡0) ∈ 푈훿. Since [푡0, 푡] ⊆ 푉 (as 푡 ∉ 푆), 훾([푡0, 푡]) ⊆ 푈 so the path

̂훾(푠) 푠 ≤ 푡0 푠 ↦ { −1 푝훿 ∘ 훾 푡0 ≤ 푠 ≤ 푡 is a lift of 훾|[0,푡] so 푡 ∈ 푆. Contradiction.

Lemma 2.3. If 푋 is path-connected the 푝 is 푛-sheeted for some 푛 ∈ N∪{∞}.

−1 Proof. Let 푥, 푦 ∈ 푋 and 훼 a path between them. Let ̂푥∈ 푝 (푥) and let ̂훼 ̂푥 be the unique lift of 훼 at .̂푥 Define a map 푝−1(푥) → 푝−1(푦)

̂푥↦ ̂훼 ̂푥(1) Now replacing 훼 with 훼 defines an inverse to this map.

Definition (degree of covering map). 푛 is called the degree of 푝.

Lemma 2.4 (homotopy lifting lemma). Let 푓0 ∶ 푌 → 푋 be a map where 푌 is path-connected. Let 퐹 ∶ 푌 × 퐼 → 푋 be a homotopy with 퐹 (⋅, 0) = 푓0. Let ̂ ̂ ̂ ̂ ̂ 푓0 ∶ 푌 → 푋 be a lift of 푓0 to 푋. Then there is a unique lift 퐹 of 퐹 to 푋 such ̂ ̂ that 퐹 (⋅, 0) = 푓0. 푦 ∈ 푌 훾 (푡) = 퐹 (푦 , 푡) Proof. Let 0 . Let 푦0 0 be a path. By path lifting lemma, ̂훾 ̂훾 (0) = 푓 ̂ (푦 ) 퐹̂ (푦 , 푡) = ̂훾 (푡) there is a unique lift 푦0 such that 푦0 0 0 such that 0 푦0 . By uniqueness of path lifting, this is the only choice for 퐹,̂ but it is not clear that 퐹̂ is continuous. We will construct a map that is obviously continuous and argue that it is also a lift. Fix 푦0. For all 푡 there exists 푈푡 an evenly covered neighbourhood of 퐹 (푦0, 푡). By definition of product topology,

−1 (푦0, 푡) ∈ 푉푡 × 퐽푡 ⊆ 퐹 (푈푡).

Compactnss of 퐼 implies that {푦0} × 퐼 is covered by 푉1 × 퐽1, … , 푉푛 × 퐽푛 where 푡 ∈ 퐽 푉 = ⋂푛 푉 푖 푖. Setting 푖=1 푖 (and passing to a path-connected subset), we have ̃ {푦0} × 퐼 covered by 푉 × 퐽1, … , 푉 × 퐽푛. Now define 퐹 on 푉 × 퐼 by

퐹̃ (푦, 푡) = 푝−1 ∘ 퐹 (푦, 푡) 훿푖 ̃ for 푦 ∈ 푉 , 푡 ∈ 퐽푖. Need to check that 퐹 is well-defined. Suppose 푡 ∈ 퐽푖 ∩ 퐽푗. Let 푦 ∈ 푉. Choose 훼 in 푉 from 푦 to 푦 and let 훼 (푠) = 퐹 (훼(푠), 푡). Now 푝−1 ∘ 훼 is 0 푡 훿푖 푡 the lift of 훼 at 퐹̂ (푦 , 푡). Same for 푝−1 ∘ 훼 so they are equal. Therefore their 푡 0 훿푗 푡 endpoints coincide: 푝−1 ∘ 퐹 (푦, 푡) = 푝−1 ∘ 퐹 (푦, 푡) . Thus 퐹̃ is well-defined. 훿푖 훿푗 퐹̃ is clearly continuous and a lift of 퐹, so it remains to check that 퐹̃ = 퐹̂ on ̃ ̂ ̃ 푉 × 퐼. By construction 퐹 (푦0, 0) = 퐹 (푦0, 0). Now 퐹 (훼(⋅), 0) is a lift of 푓0 ∘ 훼, so

10 2 Covering spaces

̂ ̃ ̂ ̃ will agree with 푓0 ∘ 훼. So 퐹 (푦, 0) = 푓0(푦) for all 푦 ∈ 푉. Finally 퐹 (푦, ⋅) is a lift ̂ ̃ ̂ of 훾푦 starting at 푓0(푦), so by uniqueness again, 퐹 (푦, 푡) =푦 ̃훾 (푡) = 퐹 (푦, 푡) for all 푦 ∈ 푉 , 푡 ∈ 퐼. We have discussed lifts of maps, paths and homotopies. Recall that homo- topy of paths is a slightly stronger form of homotopy and the next lemma shows that indeed the lift of a homotopy of paths is a homotopy of paths:

Lemma 2.5. Let 퐹 ∶ 퐼 × 퐼 → 푋 be a homotopy of paths and 퐹̂ be a lift of 퐹 to 푋̂. Then 퐹̂ is also a homotopy of paths. ̂ Proof. As 퐹 is a homotopy of path, 퐹 (0, 푡) = 푥0 for all 푡. Consider 퐹 (0, ⋅) ∶ 퐼 → 푋̂. For any 푡 ∈ 퐼 we have ̂ −1 −1 퐹 (0, 푡) ∈ 푝 (퐹 (0, 푡)) = 푝 (푥0) which is discrete. As 퐼 is connected 퐹̂ (0, … ) is constant. Same for 퐹̂ (1, … ) so 퐹̂ is a homotopy of paths.

2.3 Applications to calculations of fundamental groups

Lemma 2.6. If 푝 ∶ 푋̂ → 푋 is a map, 푥 ∈ 푋 and ̂푥∈ 푝−1(푥) then ̂ 푝∗ ∶ 휋1(푋, ̂푥)→ 휋1(푋, 푥)

is an injection.

Proof. Suppose [ ̂훾] ∈ ker 푝∗, i.e. 푝∗([ ̂훾]) = [푝 ∘ ̂훾] = [훾] = 1 ∈ 휋1(푋, 푥). Then 훾 is homotopic to the constant path. But by homotopy lifting lemma this lifts to homotopy between ̂훾 and constant path.

−1 As last time, path lifting defines an action of 휋1(푋, 푥) on 푝 (푥) by −1 −1 휋1(푋, 푥) × 푝 (푥) → 푝 (푥) ([훾], ̂푥)↦ ̂푥.훾 where ̂푥.훾 is the endpoint of the lift of 훾 at .̂푥 Note that by Lemma 2.5 this is indeed in the fibre of 푥. Furthermore it shows that this is well-defined. Finally note that this is a right action (ultimately because we defined concatenation of paths from left to right). Given 퐺 action on 푋, orbit-stabiliser says that there is a bijection between 푥 the left cosets of stabiliser 퐺푥 of an element 푥 and the orbit 퐺 . Furthermore, 퐺 has a natural action on the left cosets 퐺/퐺푥 such that the bijection is 퐺- equivariant. Spelling this out (and use right action instead of left), we have

Lemma 2.7. Suppose 푋̂ is path-connected and 푥 ∈ 푋. Let ̂푥∈ 푝−1(푥). Then

̂ −1 푝∗휋1(푋, ̂푥)\휋1(푋, 푥) → 푝 (푥) ̂ (푝∗휋1(푋, ̂푥))[훾] ↦ ̂푥.훾

11 2 Covering spaces

Furthermore, the map is equivariant. Proof. Suffices to show that the action is transitive and the stabiliser of ̂푥 is ̂ ̂ 푝∗휋1(푋, ̂푥). As 푋 is path-connected there exists a path ̂훾 between any two points in 푝−1(푥), whose image 훾 under 푝 is a loop bases at 푥, and is the only loop whose lift is ̂훾 by uniquenss. The stabiliser of ̂푥 are precisely the homotopy classes of loops based at 푥 whose lifts are loops baesd at ,̂푥 which is precisely ̂ 푝∗휋1(푋, ̂푥).

Definition (universal cover). If 푝 ∶ 푋̃ → 푋 is a covering map with 푋 path- connected and 푋̃ simply connected then 푋̃ is called a universal cover of 푋.

Corollary 2.8. If 푝 ∶ 푋̃ → 푋 is a universal cover and 푝( ̃푥)= 푥 then

−1 휋1(푋, 푥) → 푝 (푥) [훾] ↦ ̃푥.훾

is an equivariant bijection. The map is not only bijective, but also equivariantly so. Thus by looking into the universal cover we can recover information about the fundamental group of the base space. Example (fundamental group of 푆1). Consider 푝 ∶ R → 푆1, 푡 ↦ 푒2휋푖푡 is a covering map. Since R is contractible, this is the universal cover so 1 −1 휋1(푆 , 1) → 푝 (1) = Z [훾] ↦ 0.훾 is a bijection. Therefore we can write down representative loops for each element 1 1 of 휋1(푆 , 1). For 푛 ∈ Z, let 푛̃훾 (푡) = 푛푡 so 훾푛 = 푝 ∘푛 ̃훾 is a loop in 푆 based at 1. 1 As [훾푛] ↦ 푛, these represent every element of 휋1(푆 , 1) uniquely. To recover the group structure, note that for any 푚, 푛 ∈ Z, 푚 +푛 ̃훾 is the lift of 훾푛 at 푚. On the other hand, the endpoint of the lift of 훾푚 ⋅ 훾푛 at 0 is 푚 + 푛, which is the endpoint of 푚 +푛 ̃훾 . So

푚 + 푛 ∶ [훾푚 ⋅ 훾푛] ↦ 푚 + 푛 is a homomorphism. Thus 1 휋1(푆 , 1) ≅ Z.

2.4 The fundamental group of 푆1

2 1 2 Theorem 2.9. id푆1 does not extend over 퐷 , i.e. 푆 is not a retract of 퐷 .

2 1 Proof. Suppose otherwise and 푟 ∶ 퐷 → 푆 is a retraction. Then id푆1 = 푟 ∘ 푖:

푆1 id 푆1 푟 푖 퐷2

12 2 Covering spaces

Look at the induced fundamental groups, we have = 푟 ∘ 푖 idZ ∗ ∗ so id Z Z

푟∗ 푖∗ 0 Absurd.

Corollary 2.10 (Brouwer fixed point theorem). Every continuous map 푓 ∶ 퐷2 → 퐷2 has a fixed point. Proof. If there exists 푓 such that 푓(푥) ≠ 푥 for all 푥 ≠ 퐷2 then we can construct a continuous retraction 푟 ∶ 퐷2 → 푆1: for all 푥 ∈ 퐷2, let 푟(푥) be the intersection of the ray from 푓(푥) to 푥 with 푆1 (well-defined since 푓(푥) ≠ 푥). It is continuous. As 푟 fixes 푆1 this is a retract.

Theorem 2.11 (fundamental theorem of algebra). Every nonconstant poly- nomial 푝 ∶ C → C has a root. 푑 푑−1 Sketch of proof. Suppose 푝(푧) = 푧 +푎푑−1푧 +⋯+푎1푧 +푎0 has no root. Then 푝 ∶ C \ {0} → C \ {0}. Let 푟 ∶ C \ {0} → 푆1 푧 푧 ↦ |푧| be the usual retraction. Let 휆푅(푧) = 푅푧 for 푅 > 0 and consider 푓푅 which is the composition 휆푅 푝 푟 푆1 C \ {0} C \ {0} 푆1 as all these maps are homotopic, they induce the same map 푓∗ ∶ Z → Z which is multiplication by some number 푚, independent of 푅. When 푅 is small, we can argue that 푓푅 is homotopic to a constant map so 푚 = 0. When 푅 is large, 푝 is approximately 푧 ↦ 푧푑 so 푚 = 푑, contradiction.

2.5 Existence of universal covers

Theorem 2.12. If 푋 is path-connected and locally simply connected then 푋 has a universal cover. Sketch of proof. [non-examinable] Let

픛 = {훾 ∶ 퐼 → 푋 ∶ 훾(0) = 푥0} and define 푋̃ = 픛/ ≃, the homotopy classes of paths. Define 푝 ∶ 푋̃ → 푋 [훾] ↦ 훾(1) The verification is omitted.

13 2 Covering spaces

2.6 The Galois correspondence

Definition (covering space isomorphism). Let 푋 be a path-connected topo- ̂ ̂ logical space and 푝1 ∶ 푋1 → 푋, 푝2 ∶ 푋2 → 푋 are covering spaces of 푋. An ̂ ̂ isomorphism of covering spaces is a map 휑 ∶ 푋1 → 푋2 such that 푝2 ∘ 휑 = 푝1. If 1̂푥 ,2 ̂푥 are bases points and 휑(1 ̂푥 ) =2 ̂푥 , we say 휑 is based. ̂ Remark. 휑 is a lift of 푝1 to 푋2.

Theorem 2.13 (Galois correspondence with base points). Let 푋 be path- connected, locally simply connected space and 푥0 ∈ 푋. Then there is a bijection between based isomorphism class of path-connected covering space ̂ 푝 ∶ (푋,0 ̂푥 ) → (푋, 푥0) and subgroups of 휋1(푋, 푥0), given by ̂ ̂ 푋 ↦ 푝∗휋1(푋,0 ̂푥 ).

Proof. Non-examinable and omitted.

Example. Let 푋 = 푆1, we have path-connteced covering space 푝 ∶ R → 푆1, 푡 ↦ 2휋푖푡 1 1 푛 푒 and 푝푛 ∶ 푆 → 푆 , 푧 ↦ 푧 . The subgroups of Z are precisely 푛Z. It is easy to see that 푝 corresponds to 0 and 푝푛 correponds to 푛Z. Galois correspondence then tells us that these are all the path-connected covering space of 푆1 up to isomorphism.

Corollary 2.14. Let 푋 be “reasonable”. Then any two universal covers ̃ ̃ 푝1 ∶ 푋1 → 푋, 푝2 ∶ 푋2 → 푋 are isomorphic.

Proof. Exercise.

Corollary 2.15. Let 푋 be path-connected, locally simply connected and 푥0 ∈ 푋. Then there is a bijection between isomorphism class of path- ̂ connected covering space 푝 ∶ (푋,0 ̂푥 ) → (푋, 푥0) and subgroups of 휋1(푋, 푥0) modulo conjugation, given by ̂ ̂ 푋 ↦ 푝∗휋1(푋,0 ̂푥 ).

Proof. Surjectivity of the map follows from immediately from the previous the- ̂ ̂ orem. We need to prove that if 푝1∗휋1(푋1,1 ̂푥 ) and 푝2∗휋1(푋2,2 ̂푥 ) are conjugate ̂ ̂ then 푋1 and 푋2 are isomorphic covering spaces. So let ̂ ̂ 푝1∗휋1(푋1,1 ̂푥 ) = [훾]푝2∗휋1(푋2,2 ̂푥 )[훾]. (∗)

′ Let ̂훾 be the lift of 훾 and 2̂푥 = ̂훾(1).(∗) then tells us that

̂ ̂ ̂ ′ 휕1∗휋1(푋1,1 ̂푥 ) = 푝2∗ #̂훾 휋1(푋2,2 ̂푥 ) = 푝2∗휋1(푋2,2 ̂푥 ).

Then by the original Galois correspondence, there is a based isomorphism be- ̂ ̂ tween 푋1 and 푋2. Of course they are isomorphic.

14 2 Covering spaces

Definition (covering transformation). Let 푝 ∶ 푋̂ → 푋 be a covering space. A covering transformation or deck transformation 푋̂ → 푋̂ is a homeomor- phism that is also a cover isomorphism.

Corollary 2.16. Let 푋 be “reasonable”, path-connected and locally simply ̃ −1 connected and 푝 ∶ 푋 → 푋 a universal cover. Let 푥0 ∈ 푋 and 0̃푥 ∈ 푝 (푥0). −1 ̃ Let ̃푥∈ 푝 (푥0). Then there is a unique covering transformation 휑 ̃푥 ∶ 푋 → ̃ 푋 such that 휑 ̃푥(0 ̃푥 ) =. ̃푥 ̃ ̃ Proof. Both (푋,0 ̃푥 ) and (푋, ̃푥) correspond ot the trivial subgroup of 휋1(푋, 푥0) so the result follows from 2.27.

Now we have two different correspondences: Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor. Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis accumsan semper. In fact these are isomorphic. automorphism of universal covers is isomorphic to fundamental group of base group. ̃ We can thus make 휋1(푋, 푥0) act on 푋 on the left by covering transformation.

Remark. Left vs. right action. Abelian group in case of 푆1.

15 3 Seifert-van Kampen theorem

3 Seifert-van Kampen theorem

So far we have only seen one space with nontrivial fundamental group. In gen- eral, the fundamental groups are notoriously difficult to compute. In this chap- ter, we will develop the machinery needed to divide and conquer the problem of finding the fundamental group of a complex space. Specifically, given 푋 = 푌1 ∪ 푌2, we will ultimate describe 휋1푋 in terms of 휋1푌1, 휋1푌2 and 휋1(푌1 ∩ 푌2). But before that, we have to develop more group theory.

3.1 Free groups and presentations We have seen groups described in the following form in IA Groups:

2 푛 −1 퐷2푛 = ⟨푟, 푠|푠 = 푟 = 푒, 푠푟푠 = 푟 ⟩ where we impose relations on the right on the group generated by the generators on the left. This is an example of a presentation. What should be the group generated by the generators be? Should it, for example, have an elemnet of order 2? Morally, the answer should be “no” as we should move all relations to the right. This leaves us with a free group, which is a group with no relation. Given a set 퐴 of generators, called an alphabet, 퐹 퐴 is the free group generated by 퐴. Thus a free group has presentation

퐹 퐴 = ⟨푎 ∈ 퐴⟩.

Formally

Definition (free group). A group 퐹 (퐴) equipped with a map of set 퐴 → 퐹 (퐴) is the free group on 퐴 if it satisfies the following universal property: whenever 퐺 is a group and 퐴 → 퐺 is a set map there is a unique canonical homomorphism 푓 ∶ 퐹 (퐴) → 퐺 such that

퐹 (퐴) 푓

퐴 퐺

commutes. Example. 1. 퐹 (∅) ≅ 1.

2. Let 퐴 = {푎}. If 퐴 → 퐺, 푎 ↦ 푔, define 푓 ∶ Z → 퐺, 푛 ↦ 푔푛. Then the diagram Z 푓

퐴 퐺

commutes. Thus Z is the free group on 퐴. Remark.

16 3 Seifert-van Kampen theorem

1. Free group is defined uniquely up to a unique isomorphism: suppose 퐴 → 퐹 ′(퐴) also satisfies the universal property. Take 퐺 = 퐹 ′(퐴) in the universal property for 퐹 (퐴), then there is a canonical homomorphism 푓 ∶ 퐹 (퐴) → 퐹 ′(퐴) such that

퐹 (퐴)

퐴 퐹 ′(퐴)

commutes. Conversely, take 퐺 = 퐹 (퐴) in the universal property for 퐹 ′(퐴), then there is a canonical homomorphism 푓′ ∶ 퐹 ′(퐴) → 퐹 (퐴) such that the corresponding diagram commutes. Now both id퐹(퐴) and ′ ′ 푓 ∘ 푓 both make the diagram commute so by uniqueness 푓 ∘ 푓 = id퐹(퐴). ′ ′ Likewise 푓 ∘ 푓 = id퐹 ′(퐴) so 푓 and 푓 are isomorphisms. 2. The definition does not guarantee the existence of free groups. We’ll cover this later. Notation. We identify 푎 ∈ 퐴 with its image in 퐹 (퐴).

Definition (presentation). Let 퐴 be an alphabet. A subset 푅 ⊆ 퐹 (퐴) defines a (group) presentation

⟨퐴|푅⟩ = 퐹 (퐴)/⟨⟨푅⟩⟩

where ⟨⟨푅⟩⟩ is the normal closure of 푅 in 퐹 (퐴).

Example. 1. ⟨푎|푎푛⟩ ≅ Z/푛Z. 푛 2 2. ⟨푟, 푠|푟 , 푠 , 푠푟푠푟⟩ ≅ 퐷2푛.

Lemma 3.1 (universal property of group presentation). Given a presenta- tion ⟨퐴|푅⟩ and the quotient map 푞 ∶ 퐹 (퐴) → ⟨퐴|푅⟩, for any homomorphism 푔 ∶ 퐹 (퐴) → 퐺 such that 푔(푟) = 1 for all 푟 ∈ 푅, there exists a unique homo- morphism 푓 ∶ ⟨퐴|푅⟩ → 퐺 such that 푓 ∘ 푞 = 푔. In other words, the following diagram commutes: ⟨퐴|푅⟩

푞 푓 푔 퐹 (퐴) 퐺

Proof. Follows easily from universal property of quotient map.

Definition (pushout). Let 푖 ∶ 퐶 → 퐴, 푗 ∶ 퐶 → 퐵 be group homomorphisms. Homomorphism 푘 ∶ 퐴 → Γ, ℓ ∶ 퐵 → Γ is a pushout if it satisfies the following property: for any group 퐺 and homomorphisms 푓 ∶ 퐴 → 퐺, 푔 ∶ 퐵 → 퐺 such that 푓 ∘ 푖 = 푔 ∘ 푗, then there is a unique homomorphism 휑 ∶ Γ → 퐺 such that

17 3 Seifert-van Kampen theorem

푓=휑∘푘,푔=휑∘ℓ. In other words the following diagram commutes.

퐶 푖 퐴

푗 푘 푓 퐵 ℓ Γ 휑 푔 퐺

Again Γ is uniquely defined by the universal property. We mainly care about special cases of the definition.

Definition (free product, amalgamated free product). If 퐶 ≅ 1, then Γ is called the free product of 퐴 and 퐵, denoted 퐴 ∗ 퐵. More generally, if 푖 and 푗 are injective then Γ is called the amalgamated free product, denoted 퐴 ∗퐶 퐵.

Example. Z ∗ Z ≅ 퐹2 since they satisfy the same universal property. More generally, we can check that

Z⏟⏟⏟⏟⏟∗ Z ∗ ⋯ ∗ Z ≅ 퐹푟. 푟

Notation. Write 퐹푛 for the free group with 푛 generators.

Lemma 3.2. 퐶 푖 퐴

1 퐴/⟨⟨푖(퐶)⟩⟩

is a pushout.

Proof. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor. Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis accumsan semper. presentation for free group with amalgamation

3.2 Seifert-van Kampen theorem for wedges

18 3 Seifert-van Kampen theorem

Definition (wedge). Given two pointed spaces (푋, 푥0), (푌 , 푦0), the wedge is 푋 ∨ 푌 = 푋 ⨿ 푌 /(푥0 ∼ 푦0).

Usually 푋 and 푌 are path-connected so we can define wedges 푋 ∨ 푌 without specifying basepoints.

Theorem 3.3 (Seifert-van Kampen for wedges). If 푌1, 푌2 are path-connected and 푥0 is the wedge point of 푋 = 푌1 ∨ 푌2. Then

휋1(푋, 푥0) = 휋1(푌1, 푥0) ∗ 휋1(푌2, 푥0).

Sketch of proof. non-examinable Suppose 푓1 ∶ 휋1(푌푖, 푥0) → 퐺 are group homomorphisms for 푖 = 1, 2. We need to find a unique 휙 ∶ 휋1(푋, 푥0) → 퐺 such that 휙 restricts to 푓푖 on 휋1(푌푖, 푥0). First replace 푋 by 푋′ (drawing) with 푋 ≃ 푋′. Let 훾 ∶ 퐼 → 푋′ be a based loop. We can “straighten” 훾 so that it is of the form

훾 = 훼1 ⋅ 훽1 ⋅ 훼2 ⋅ 훽2 ⋯ 훼푛 ⋅ 훽푛 where 훼푖’s are in 휋1(푌1, 푥0) and 훽푖’s are in 휋1(푌1, 푥0). Define

휙(훾) = 푓1(훼1)푓2(훽2)푓1(훼2) ⋯ 푓2(훽푛−1)푓1(훼푛)푓2(훽푛) uniquely. This is easily seen to be a homomorphism but we need to prove that ′ 휙 is well-defined. Let 훾 ≃퐹 훾 with

′ ′ ′ ′ ′ 훾 = 훼1 ⋅ 훽1 ⋯ 훼푚훽푚 so ′ ′ ′ ′ ′ 휙(훾 ) = 푓1(훼1)푓2(훽1) ⋯ 푓1(훼푚)푓2(훽푚), we need to prove that 휙(훾) = 휙(훾′). The key idea is to “straighten” 퐹 so that it −1 is “transverse” to 푥0: this means that 퐹 (푥0) ⊆ 퐼 × 퐼 consists of a finite union of circles and intervales embedded in 퐼 × 퐼. If there is a cirlce 푆! ⊆ 퐹 −1(0) then we can “cut it out” and remove it. An arc with both endpoints on 훾 exhibit a 훿 ⊆ Γ 훿 ≃ 푐 푌 푌 푛 휙(훾) subarc such that 푥0 in 1 or 2, reducing without changing . After finitely many of these moves, we are left with a picture of the following ′ ′ ′ form (drawing). Therefore 푚 = 푛 and 훼푖 ≃ 훼푖, 훽푖 ≅ 훽푖 as paths so 휙(훾) = 휙(훾 ) as required.

Example. Let 푋 = 푆1 ∨ 푆1, then

1 1 휋1푋 ≅ 휋1푆 ∗ 휋1푆 ≅ Z ∗ Z ≅ 퐹2. 푋 = ⋁푟 푆1 More generally, let 푟 푖=1 , sometimes called a bouquet, then

휋1푋푟 ≅ Z⏟∗ ⋅ ∗ Z ≅ 퐹푟. 푟

3.3 Seifert-van Kampen theorem

19 3 Seifert-van Kampen theorem

Theorem 3.4 (Seifert-van Kampen). If 푋 = 푌1 ∪푍 푌2 with 푌1, 푌2, 푍 open and path-connected and 푥0 ∈ 푍 then the diagram

푖1∗ 휋1(푍, 푥0) 휋1(푌1, 푥0)

푖2∗ 푗1∗

푗2∗ 휋1(푌2, 푥0) 휋1(푋, 푥0)

is a pushout.

Proof. Omitted.

푛 Example. Let 푋 = 푆 where 푛 ≥ 2. Let 푥± = (±1, 0, … , 0) be the north/south poles and define

푛 푈± = 푆 − {푥∓} 푛 푉 = 푈+ ∩ 푈− = 푆 − {푥±}

푛 Then 푋 = 푈+ ∪푉 푈−. Stereographic projection tells us that 푈± ≅ R . Project 푉 radially onto the cylinder (−1, 1) × 푆푛−1, which is a homeomorphism so 푉 ≅ (−1, 1) × 푆푛−1 ≃ 푆푛−1. 푆푛−1 is path-connected for 푛 ≥ 2 so by Seifert-van Kampen the following diagram is a pushout:

푛−1 휋1(푆 , 푥0) 1

푛 1 휋1(푆 , 푥0)

푛 so 휋1(푆 , 푥0) is a quotient of 1 so is trivial.

Definition (neighbourhood deformation retract). A subset 푌 ⊆ 푋 is called a neighbourhood deformation retract if there exists 푌 ⊆ 푉 ⊆ 푋 where 푉 is open in 푋 such that 푌 is a deformation rectraction of 푉.

Corollary 3.5. If 푋 = 푌1 ∪푍 푌2 with 푌1, 푌2, 푍 path-connected and closed and 푍 a neighbourhood deformation retract of 푌1 and 푌2 and 푥0 ∈ 푍 then

휋1(푍, 푥0) 휋1(푌1, 푥0)

휋1(푌2, 푥0) 휋1(푋, 푥0)

is a pushout.

Proof. See online notes.

3.4 Attaching cells

20 3 Seifert-van Kampen theorem

Definition (cell). An 푛-cell is a copy of 퐷푛, the closed ball in R푛.

Definition. Let 훼 ∶ 푆푛−1 = 휕퐷푛 → 푋 be a continuous map. The space

푛 푛 푋 ∪훼 퐷 ∶= 푋 ⨿ 퐷 / ∼

where ∼ is the finest equivalence relation such that 훼(휃) ∼ 휃 for all 휃 ∈ 푆푛−1, is called an attaching cell.

What effect does attaching an 푛-cell have on 휋1? Let’s start with 푛 ≥ 3:

Lemma 3.6. If 푛 ≥ 3 and 훼 ∶ 푆푛−1 → 푋 is a continuous map. Let 푛−1 푥0 = 훼(휃0) for 휃0 ∈ 푆 . Then the (not necessarily injective) inclusion map 푛 푛 푖 ∶ 푋 → 푋 ∪훼 퐷 induces an isomorphism 푖∗ ∶ 휋1(푋, 푥0) → 휋1(푋 ∪훼 퐷 , 푥0).

Proof. The main obstacle is that 훼 might not be injective. However, we can divide 퐷푛 into two parts and attach 퐷푛 in two stages: the mapping cylinder of 훼 is 푛−1 푀훼 ∶= 푋 ⨿ (푆 × 퐼)/ ∼ where 훼(휃) ∼ (휃, 0) for all 휃 ∈ 푆푛−1. Note that

1. 푋 is a deformation retract of 푀훼. 푛−1 2. 푆 × {1} ⊆ 푀훼 is a neighbourhood deformation retract. 3. 푆푛−1 ⊆ 퐷푛 is a neighbourhood deformation retract.

푛−1 If we choose 휃1 ∈ 푆 , the previous corollary tells us that

푛−1 휋1(푆 , 휃1) 휋1(푀훼, 휃1)

푗∗ 푛 푛 휋1(퐷 , 휃1) 휋1(푀훼 ∪푆푛−1 퐷 , 휃1)

푛 is a pushout. Therefore the inclusion 푗 ∶ 푀훼 → 푀훼 ∪푆푛−1 퐷 induces an 푛 푛 ′ isomorphism on 휋1. Since 푋 ∪훼 퐷 = 푀훼 ∪푆푛−1 퐷 and 푀훼 deformation retracts to 푋, the result follows. What about 푛 = 2?

1 Lemma 3.7. If 훼 ∶ 푆 → 푋 is a continuous map and 푥0 = 훼(휃0) where 1 휃0 ∈ 푆 . Then 2 휋1(푋 ∪훼 퐷 , 휃0) ≅ 휋1(푋, 푥0)/⟨⟨[훼]⟩⟩ 푋 ↪ 푋 ⋂ 퐷2 and the inclusion map 훼 induces the quotient map

2 휋1(푋, 푥0) → 휋1(푋 ∪훼 퐷 , 푥0).

21 3 Seifert-van Kampen theorem

Proof. As in the proof of the previous lemma, the diagram

1 훼∗ 휋1(푆 , 휃0) 휋1(푋, 푥0)

푖∗

2 2 휋1(퐷 , 휃0) 휋1(푋 ∪훼 퐷 , 푥0) is a pushout. By lemma 3.2 the result follows.

Theorem 3.8. If 퐺 = ⟨퐴|푅⟩ with 퐴, 푅 both finite then it is the fundamental group of some space. Moreover the spaces can be taken to be compact.

In fact, we don’t have to restrict our attention to finitely generated or finitely presented groups. So every group is the fundamental group of some space (al- though not compact in general).

Proof. If 푅 = {푟1, … 푟푛} then

퐺 = 퐹 (퐴)/⟨⟨푟1, … , 푟푛⟩⟩

≅ (퐹 (퐴)/⟨⟨푟1, … , 푟푛−1⟩⟩)/⟨⟨푟푛⟩⟩ ≅ …

≅ (… (퐹 (퐴)/⟨⟨푟1⟩⟩) … )/⟨⟨푟푛⟩⟩, one way to check this is to show they satisfy the same universal property. Now induciton on 푛, with the base case 푛 = 1 being the wedge of |퐴| circles.

3.5 Classification of surfaces

Definition (topological manifold). An 푛-dimensional (topological) manifold is a Hausdorff space 푀 such that every 푥 ∈ 푀 has an open neighbourhood 푈 homeomorphic to an open subset of R푛.

Definition (surface). A 2-dimensional manifold is called a surface.

1 2 2 2 Example. Let 훼 ∶ 푆 → ∗. Consider 푋 = ∗ ∪훼 퐷 . Note that ∫ 퐷 ≅ R and 2 2 푆 − {푥+} ≅ R via stereographic projection. Moreover the homeomorphism 2 1 2 푖 ∶ ∫ 퐷 → 푆 − {푥+} extends to a unique continuous bijection 푋 → 푆 , so a homeomorphism. In particular 푆2 is a surface. Γ = ⋁2푔 푆1 푆′ ≅ 푆1 Example. Let 2푔 푖=1 푖 , with each 푖 . Choose unit speed loops 훼1, … , 훼푔 and 훽1, … , 훽푔 in the circles. Let

휌푔 = (훼1 ⋅ 훽1 ⋅ 훼1 ⋅ 훽1) ⋅ (훼2 ⋅ 훽2 ⋅ 훼2 ⋅ 훽2) … (훼푔 ⋅ 훽푔 ⋅ 훼푔 ⋅ 훽푔) and let Σ = Γ ∪ 퐷2. 푔 2푔 휌푔

Claim Σ푔 is a surface. There are three cases to consider. If a point in the interior of 퐷2 then it has a neighbourhood homeomorphic to an open disk. If a

22 3 Seifert-van Kampen theorem point is in the interior of image of a path the “two parts” glue together to form an open disk. Similary all the edges are identified together. 2 Σ0 is just 푆 . Σ1 is the square with two sides identified to a torus. In general Σ푔 is called the (orientable surface) with genus 푔. Γ = ⋁푔 푆1 Example. Let 푔+1 푖=0 푖 and let

휎푗 = 훼0 ⋅ 훼0 ⋅ 훼1 ⋅ 훼1 … 훼푔 ⋅ 훼푔 let 푆 = Γ ∪ 퐷2. 푔 푔+1 휎푔 Similarly we can check these are surfaces. This is the non-orientable surface of 2 genus 푔. 푆0 = R푃 and 푆1 is the Klein bottle. We have

−1 −1 −1 −1 휋1Σ푔 = ⟨푎1, … , 푎푔, 푏1, … , 푏푔|푎1푏1푎1 푏1 ⋯ 푎푔푏푔푎푔 푏푔 ⟩ 2 2 2 휋1푆푔 = ⟨푎0, … , 푎푔|푎0푎1 ⋯ 푎푔⟩

We state without proof

Theorem 3.9 (classification of compact surfaces). If 푀 is a compact surface then either 푀 ≅ Σ푔 or 푀 ≅ 푆푔.

We won’t prove this but a good point to start is to consider given an iden- tification of 푆1 of 퐷2, how can we convert it into one of the two forms? We also ask the following question: are {Σ푔} and {푆푔} pairwise non-homeomorphic? What about homotopy equivalence? The only tool available to us is 휋1. The strategy is to that the fundamental groups map onto different abelian groups.

Lemma 3.10. Let 푔 ∈ N. 2푔 2푔 1. The group 휋1Σ푔 surjects Z but not Z ⊕ (Z/(2)).

푔 푔+1 2. The group 휋1푆푔 surjects Z ⊕ (Z/(2)) but not Z .

Proof. Easy. See notes. And as a result we get want we want

Corollary 3.11. The strategy works.

23 4 Simplicial complexes

4 Simplicial complexes

We have seen that the fundamental groups are useful, and for example, when it works, it tells us 푆푛 is contractible for 푛 > 1. There are higher dimensional analogues of 휋1, called the homotopy groups 휋푛. However, they are notoriously difficult to compute. Instead, we will use (more or less) the only thing inmathe- matics we understand fully (again, more or less) — linear algebra. This is called homology. There are many types of homologies and we’ll only define simplicial homolo- gies in this course.

4.1 Simplices and stuff

Definition. A finite set 푉 ⊆ R푛 is in general position if the smallest affine subspace containing 푉 is of dimension |푉 | − 1.

This is quite an abstract definition, but there are a few equivalent notions. 푉 = {푣 , … , 푣 } 푡 , … 푡 ∑푛 푡 = 0 For example, if 0 푛 then for any 0 푛 such that 푖=0 푖 , if ∑푛 푡 푣 = 0 푡 = 0 푖 푖=0 푖 푖 then 푖 for all .

푚 Definition (simplex). For 푛 ≥ 0, 푉 = {푣0, … , 푣푛} ⊆ R . The span of 푉 is 푛 푛

⟨푉 ⟩ = {∑ 푡푖푣푖 ∶ 푡푖 ≥ 0, ∑ 푡푖 = 1}. 푖=0 푖=0

If 푉 is in general position, 휎 = ⟨푉 ⟩ is an 푛-simplex.

Definition (face). Let 푉 = {푣0, … , 푣푛} in general position. If 푈 ⊆ 푉 then ⟨푈⟩ is called a face of ⟨푉 ⟩, write ⟨푈⟩ ≤ ⟨푉 ⟩. If 푈 ≠ 푉 then ⟨푈⟩ is called a proper face.

Definition (simplicial complex, dimension, skeleton). A simplicial complex is a finite set of simplices 퐾 in some R푚 satisfying the following condition: 1. if 휎 ∈ 퐾 and 휏 ≤ 휎 then 휏 ∈ 퐾,

2. if 휎, 휏 ∈ 퐾 then 휎 ∩ 휏 ≤ 휎 and 휎 ∩ 휏 ≤ 휏. The dimension of 퐾, denoted dim 퐾, is the largest 푛 such that 퐾 contains an 푛-simplex. The 푑-skeleton of 퐾 is

퐾(푑) = {휎 ∈ 퐾 ∶ dim 휎 ≤ 푑}.

Example. 1. If 휎 is a simplex then 퐾 = {휏 ∶ 휏 ≤ 휎} is a simplicial complex. 2. If 휎 is a simplex then the set of proper faces of 휎, denoted 휕휎, is a simplicial complex. It is called the boundary of 휎. The set of points in 휎 not in a simplex of 휕휎 is called the interior, denoted by ̊휎.

24 4 Simplicial complexes

Note that if 휎 is a 0-simplex then ̊휎= 휎.

Definition (realisation/polyhedron). The realisation or polyhedron of a simplicial complex 퐾 is the union of the simplices in 퐾, denoted by |퐾|.

Example. 푛+1 1. In R , the standard basis {푒0, … 푒푛} is in general position. The simplex it spans 휎푛 = ⟨푒0, … , 푒푛⟩ is called the standard 푛-simplex.

2. The standard (simplicial) (푛 − 1)-sphere is 휕휎푛.

Definition (triangulation). A triangulation of a space 푋 is a homeomor- phism ℎ ∶ |퐾| → 푋.

푛−1 It’s not hard to see that there is a triangulation ℎ ∶ |휕휎푛| → 푆 . Example. Here is another way of triangulating 푆푛. For now set 푛 = 2. The convex hull of {±푒0, ±푒1, ±푒2} is a surface of an octahedron, which is trian- 2 푛+1 gulation of 푆 . In general, let {푒0, … , 푒푛} be the standard basis for R and 퐸 = {±푒0, … , ±푒푛}. Let

퐸0 = {푆 ⊆ 퐸 ∶ for all 푖 exactly one of ±푒푖 is in 푆}.

Let 퐾 = {⟨푆⟩ ∶ 푆 ∈ 퐸0}. This is the octahedral 푛-sphere and there exists a triangulation |퐾| → 푆푛.

Definition (simplicial map). Let 퐾, 퐿 be simplicial complexes. A simplicial map 푓 ∶ 퐾 → 퐿 is a map such that for all ⟨푣0, … , 푣푛⟩ ∈ 퐾,

푓(⟨푣0, … , 푣푛⟩) = ⟨푓(푣0), … , 푓(푣푛)⟩

where 푓({푣푖}) = {푓(푣푖)}. The realisation of 푓 ∶ 퐾 → 퐿 is the continuous map |푓| ∶ |퐾| → |퐿| defined on 휎 = ⟨푣0, … , 푣푛⟩ to be

푛 푛

푓휎 (∑ 푡푖푣푖) = ∑ 푡푖푓(푣푖). 푖=0 푖=0

Note that if 휏 ≤ 휎 then 푓휏 = 푓휎|휏, so |푓| is well-defined and continuous. Example. (drawing)

4.2 Barycentric subdivision Realisaition of simplicial maps are piecewise linear and thus very rigid. On the other hand, the realisations of simplicial complexes, as topological spaces, are “deformable”. Is every continuous map |퐾| → |퐿| homotopic to a realisation of a simplicial map? For example for 퐾 = 퐿 = 휕휎2, there are infinitely many 1 homotopy classes of continuous maps, which are in bijection with 휋1(푆 ). On the other hand there are only finitely many simplicial map 퐾 → 퐿, and thus at most that many realisations. To establish the correspondence, we need subdivision.

25 4 Simplicial complexes

Definition (barycentre). If 휎 = ⟨푣0, … , 푣푛⟩, the barycentre of 휎 is 1 푛 ̂휎푛 = ∑ 푣푖. 푛 + 1 푖=0

Definition (barycentric subdivision). Suppose 퐾 is a simplicial complex. The barycentric subdivision of 퐾 is 퐾′ with vertices { ̂휎∶ 휎 ∈ 퐾}.A ′ collection of barycentres { ̂휎0, … , ̂휎푛} spans a simplex in 퐾 whenever 휎0 ≤ 휎1 ≤ ⋯ ≤ 휎푛.

Lemma 4.1. 퐾′ is a simplicial complex and |퐾′| = |퐾|.

Proof. See online notes.

Definition. We define the 푟th barycentric subdivision to be

퐾(0) = 퐾 퐾(푟) = (퐾(푟−1))′

Definition (mesh). Let 퐾 be a simplicial complex. define the mesh of 퐾 to be mesh(퐾) = max ‖푣0 − 푣1‖2. ⟨푣0,푣1⟩∈퐾

Here the 2-norm is just taken for the sake of convenience and concreteness.

Lemma 4.2. If dim 퐾 = 푛 then 푛 푟 mesh(퐾(푟)) ≤ ( ) mesh(퐾). 푛 + 1

In particular lim mesh(퐾(푟)) = 0. 푟→∞

Proof. dim 퐾′ = dim 퐾 = 푛 so by induction it suffices to show that 푛 mesh(퐾′) ≤ mesh(퐾). 푛 + 1

A 1-simplex in 퐾′ is of the form ⟨ ̂휏, ̂휎⟩ where 휏 ≤ 휎. Note that 퐾′ is a finite set and mesh is realised by some pairs of vertices. By a bit geometric reasoning this is achieved by some vertex. We may thus assume that ̂휏= 푣0, a vertex of

26 4 Simplicial complexes

휎 = ⟨푣0, … , 푣푚⟩. Thus 1 푚 ‖ ̂휏− ̂휎‖ = ∥푣0 − ∑ 푣푖∥ 푚 + 1 푖=0 푚 1 푚 = ∥ 푣0 − ∑ 푣푖∥ 푚 + 1 푚 + 1 푖=1 1 푚 = ∥∑(푣0 − 푣푖)∥ 푚 + 1 푖=1 1 푚 ≤ ∑ 푣0 − 푣1 푚 + 1 푖=1 푚 ≤ mesh(퐾) 푚 + 1 푛 ≤ mesh(퐾) 푛 + 1

4.3 Simplicial approximation theorem

Definition (star). Let 푣 be a vertex of 퐾. The star of 푣 is

St퐾(푣) = ⋃ ̊휎 푣∈휎∈퐾

Definition (simplicial approximation). Let 휙 ∶ |퐾| → |퐿| be a continuous map. A simplicial map 푓 ∶ 퐾 → 퐿 is a simplicial approximation of 휙 if for every vertex 푣 of 퐾, 휙(St퐾(푣)) ⊆ St퐿(푓(푣)).

Lemma 4.3. If 푓 ∶ 퐾 → 퐿 is a simplicial approximation to 휙 ∶ |퐾| → |퐿| then |푓| ≃ 휙.

Proof. Suppose |퐿| ⊆ R푚 as usual. Consider the straightline homotopy 퐻 between |푓| and 휑. We will prove that 퐻 stays inside |퐿|. Let 푥 ∈ ̊휎 and let 휙(푥) ∈. ̊휏 We’ll show that 푓(휎) ≤ 휏. The result then follows because 휏 is a convex subset of 푅푚. Let 휎 = ⟨푣0, … , 푣푛⟩. For each 푖, 푥 ∈ St퐾(푣푖) so

휙(푥) ∈ 휙(St퐾(푣푖)) ⊆ St퐿(푓(푣푖)) so 푓(푣푖) is a vertex of 휏. So 푓(휎)휏 as desired.

Theorem 4.4 (simplicial approximation theorem). Let 퐾, 퐿 be simplicial complexes and 휙 ∶ |퐾| → |퐿| a continuous map. For some 푟 ∈ N there is a implicial approximation to 휙, 푓 ∶ 퐾(푟) → 퐿.

27 4 Simplicial complexes

Proof. Let −1 푈 = {휙 (St퐿(푢)) ∶ 푢 a vertex of 퐿} which is an open cover of |퐾|. By Lebesgue number lemma there is 훿 > 0 such that for all 푥 ∈ |퐾|, there exists a vertex of 퐿 such that

−1 퐵(푥, 훿) ⊆ 휙 (St퐿(푢)).

Choose 푟 large enough such that mesh(퐾(푟)) < 훿. Then for any vertex 푣 of 퐾(푟),

−1 St퐾(푟) (푣) ⊆ 퐵(푣, 훿) ⊆ 휙 (St퐿(푢)) for some 푢. Set 푓(푣) = 푢 for some such 푢. Left to check this is a simplicial map, i.e. for all 휎 ∈ 퐾(푟), 푓(휎) ∈ 퐿. But as in the proof of the previous lemma, if 푥 ∈ ̊휎 and 휙(푥) ∈ ̊휏 then 푓(휎) must be a face of 휏.

28 5 Homology

5 Homology

5.1 Simplicial homology The analogue in simplices of a path is a chain, which is a formal sum of simplices. If we interpret positive coefficient as copies of a simplex, what does it meanto have a negative simplex? To make sense of this we need the notion of oriented simplex.

Definition (orientation). Let 푉 = (푣0, … , 푣푛) be an ordered set of points 푀 in general position in R . Consider the natural action of 푆푛+1 on 푉. The subgroup 퐴푛+1 ≤ 푆푛+1 has 2 orbits on 푉, as long as 푛 ≥ 1. An orientation on 휎 = ⟨푉 ⟩ is a choice of 퐴푛+1-orbit under the action on 푉. We will abuse notation and write ⟨푣0, … , 푣푛⟩ for the simplex ⟨푣0, … , 푣푛⟩ equipped with the orientation which is the 퐴푛+1-orbit of (푣0, … , 푣푛).

Example. Let 푉 = {푣0, 푣1}. The two possible orientations are ⟨푣0, 푣1⟩ and ⟨푣1, 푣0⟩, which corresponds to “arrows going in opposite directions”.

Example. Let 푉 = {푣0, 푣1, 푣2}. There are two orientations, for exmaple ⟨푣0, 푣1, 푣2⟩ and ⟨푣2, 푣1, 푣0⟩ are two representatives.

Definition (chain). Let 퐾 be a simplicial complex. The group of 푛-chains on 퐾 is 퐶푛(퐾) = ⨁ ⟨휎⟩. 휎∈퐾,dim 휎=푛

In particular if there are no 푛-simplex (e.g. 푛 > dim 퐾 or 푛 < 0) then 퐶푛(퐾) ≅ 0. Arbitrarily choose orientations on the simplices of 퐾 and then identify −휎 with the opposite oriented simplex. Note that this arbitrary choice isn’t important — it could be realised by an automorphism of 퐶푛(퐾). Remark. Note that these groups are abelian, which is a huge advantage com- pared to homotopy groups if you actually want to do anything with them. On the other hand, it also means that there are things that a homotopy group can see but homology groups cannot.

Definition (boundary homomorphism). The (푛th) boundary homomor- phism 휕푛, usually just written as 휕, is defined by

퐶푛(퐾) → 퐶푛−1(퐾) 푛 푖 ⟨푣0, … , 푣푛⟩ ↦ ∑(−1) ⟨푣0, … ,푖 ̂푣 , … , 푣푛⟩ 푖=0

where 푖̂푣 means that the vertex 푣푖 is omitted.

Note this is well-defined.

Example. Let 휎 = ⟨푣0, 푣1⟩. Then 휕(휎) = ⟨푣1⟩ − ⟨푣0⟩.

29 5 Homology

Example. Let 휎 = ⟨푣0, 푣1, 푣2⟩. Then

휕(휎) = ⟨푣1, 푣2⟩ − ⟨푣0, 푣2⟩ + ⟨푣0, 푣1⟩

= ⟨푣1, 푣2⟩ + ⟨푣2, 푣0⟩ + ⟨푣0, 푣1⟩

Definition (cycle, boundary). Let 푛 ∈ Z. The group

푍푛(퐾) = ker 휕푛 ≤ 퐶푛(퐾)

is the group of 푛-cycles. The group 퐵푛(퐾) = im 휕푛+1 ≤ 퐶푛(퐾) is the group of 푛-boundaries.

These are analogous to loops and homotopies respectively.

Lemma 5.1. Every 푛-boundary is an 푛-cycle, i.e.

퐵푛(퐾) ≤ 푍푛(퐾),

i.e. 휕푛 ∘ 휕푛+1 = 0.

Proof. Let 휎 = ⟨푣0, … , 푣푛⟩. By definition 푛 푖 휕(휎) = ∑(−1) ⟨푣0, … ,푖 ̂푣 , … , 푣푛⟩ 푖=0 so

푖 푗 휕 ∘ 휕(휎) = ∑(−1) (−1) ⟨푣0, … ,푗 ̂푣 , … ,푖 ̂푣 , … , 푣푛⟩ 푖,푗<푖 푖 푗−1 + ∑(−1) (−1) ⟨푣0, … , 푣푖, … , 푣푗, … , 푣푛⟩ 푖,푗>푖 푖+푗 = ∑(−1) ⟨푣0, … ,푗 ̂푣 , … ,푖 ̂푣 , … 푣푛⟩ 푖,푗<푖 푖+푗 − ∑(−1) ⟨푣0, … ,푖 ̂푣 , … ,푗 ̂푣 , … , 푣푛⟩ 푖,푗>푖

Definition (homology group). The 푛th homology group of 퐾 is

퐻푛(퐾) = 푍푛(퐾)/퐵푛(퐾).

Remark. The homology we discuss in this course is simplicial homology, which has the advanatage that all the homology groups are finitely generated. Thus in principle, 퐻푛(퐾) can always be computed using linear algebra. But except in the following few demonstrative examples, as a man of culture you should avoid it as much as possible.

30 5 Homology

Example. Let 퐾 be the standard simplicial circle. The vertices of 퐾 are 푒0, 푒1, 푒2. Thus

3 퐶0(퐾) = ⟨푒0⟩ ⊕ ⟨푒1⟩ ⊕ ⟨푒2⟩ ≅ Z 3 퐶1(퐾) = ⟨푒0, 푒1⟩ ⊕ ⟨푒1, 푒2⟩ ⊕ ⟨푒2, 푒0⟩ ≅ Z

퐶푛(퐾) = 0 for 푛 > 1

There is only one interesting boundary map 휕 = 휕1 ∶ 퐶1(퐾) → 퐶0(퐾). Looking at the definitions, we can write down a matrix for 휕, in the bases we choose −1 0 1 ⎜⎛ 1 −1 0 ⎟⎞ ⎝ 0 1 −1⎠

After a bit of work we can put in Smith normal form 1 0 0 ⎜⎛0 1 0⎟⎞ ⎝0 0 0⎠

2 so im 휕1 ≅ Z (as a direct summand). ker 휕1 ≅ Z. Thus

3 2 퐻0(퐾) = 푍0(퐾)/퐵0(퐾) = 퐶0(퐾)/ im 휕1 ≅ Z /Z ≅ Z

퐻1(퐾) = 푍1(퐾)/퐵1(퐾) = ker 휕1/0 ≅ Z

퐻푛(퐾) = 0 for 푛 > 1

The fact that 퐻1(퐾) ≅ 푍 is related to intuitive observation that there is a “hole” in the simplicial complex. Contrast this with the next example. (We’ll interpret 퐻0(퐾) shortly)

Example. Let 퐿 be the standard 2-simplex 퐾 ∪ {휎2} where 휎2 = ⟨푒0, 푒1, 푒2⟩. We have (nontrivial) chain groups

퐶0(퐿) = 퐶0(퐾)

퐶1(퐿) = 퐶1(퐾)

퐶2(퐿) = ⟨휎2⟩

The boundary map 휕1 is same as before and for 휕2, which is

휕2(휎2) = ⟨푒0, 푒1⟩ + ⟨푒1, 푒2⟩ + ⟨푒2, 푒0⟩ which has a particularly simple matrix (1, 1, 1). In particular 휕2 is injective so ker 휕2 = 0. We know im 휕2 ⊆ ker 휕1 ≅ 푍. But we can see that im 휕2 is a direct summand of 퐶1(퐿) so im 휕2 = ker 휕1. Thus the homology groups are

퐻0(퐿) = 퐻0(퐾) ≅ Z

퐻1(퐿) = 푍1(퐿)/퐵1(퐿) = ker 휕1/ im 휕2 ≅ 0

퐻2(퐿) = 푍2(퐿)/퐵2(퐿) = ker 휕2/0 ≅ 0

Alas! The first homology group has been killed.

31 5 Homology

Lemma 5.2. Let 퐾 be a simplicial complex. If 푑 is the number of path components of |퐾| then 푑 퐻0(퐾) ≅ Z .

|휋0(퐾)| Proof. Let 휋0(퐾) be the set of path components of |퐾|. Let Z[휋0(퐾)] ≅ Z be the free abelian group generated by 휋0(퐾). There is a natural map

푞 ∶ 퐶0(퐾) → Z[휋0(퐾)] ⟨푣⟩ ↦ [푣] Because there is a vertex in every component of |퐾|, 푞 is surjective. Note that 퐵0(퐾) ⊆ ker 푞: 퐵0(퐾) is generated by elements ⟨푣⟩ − ⟨푢⟩ where ⟨푢, 푣⟩ is a 1- simplex of 퐾. Since ⟨푢⟩ and ⟨푣⟩ are in the same path component, 푞(⟨푣⟩−⟨푢⟩) = 0 so indeed 퐵0(퐾) ⊆ ker 푞. Because 퐻0(퐾) = 푍0(퐾)/퐵0(퐾) = 퐶0(퐾)/퐵0(퐾), 푞 descends to a map

퐻0(퐾) → Z[휋0(퐾)].

Left to check this is injective, i.e. ker 푞 ⊆ 퐵0(퐾). Note that ker 푞 is generated by terms of the form ⟨푣⟩ − ⟨푢⟩ where [푢] = [푣]. By simplicial approximation, there exists a “simplicial path” from 푢 to 푣

푐 = ⟨푣0, 푣1⟩ + ⟨푣1, 푣2⟩ + ⋯ + ⟨푣푘−1, 푣푘⟩ where 푣0 = 푢, 푣푘 = 푣. But 휕1(푐) = ⟨푣⟩ − ⟨푢⟩ ∈ 퐵0(퐾) as required.

5.2 Chain complexes & chain homotopies

Definition (chain complex). A chain complex 퐶• is a sequence of abelian (퐶 ) 퐶 = 0 푛 < 0 groups 푛 푛∈Z with 푛 for and boundary homomorphisms 휕푛 ∶ 퐶푛 → 퐶푛−1 such that 휕푛−1 ∘ 휕푛 = 0 for all 푛. A chain map 푓• ∶ 퐶• → 퐷• is a sequence of homomorphisms 푓푛 ∶ 퐶푛 → 퐷푛 such that the following diagram commutes for all 푛:

푓푛 퐶푛 퐷푛

휕푛 휕푛

푓푛−1 퐶푛−1 퐷푛−1

Note. Note that we suppress the notational distinction between the boundary homomorphisms of 퐶• and 퐷•. This is a common practice as they have different domains and there is little room for confusion.

Definition (boundary, cycle, homology). If 퐶• is a chain complex, then define boundaries 퐵푛 and cycles 푍푛

퐵푛(퐶•) = im 휕푛+1 ≤ 퐶푛

푍푛(퐶•) = ker 휕푛 ≤ 퐶푛

32 5 Homology

The 푛th homology is defined as

퐻푛(퐶•) = 푍푛(퐶•)/퐵푛(퐶•).

Lemma 5.3. A chain map 푓• ∶ 퐶• → 퐷• induces a well-defined homomor- phism 푓∗ ∶ 퐻푛(퐶•) → 퐻푛(퐷•) for all 푛.

Proof. Trivial from commutativity of 휕 and 푓푛. 퐾 (퐶 ) Example. If is a simplicial complex then 푛 푛∈Z form a chain complex 퐶•(퐾).

Lemma 5.4. A simplicial map 푓 ∶ 퐾 → 퐿 induces a chain map 푓• ∶ 퐶•(퐾) → 퐶•(퐿) by

푓푛 ∶ 퐶푛 → 퐷푛 푓(휎) if dim 푓(휎) = dim 휎 휎 ↦ { 0 if dim 푓(휎) < dim 휎

Therefore 푓 induces homomorphisms 푓∗ ∶ 퐻푛(퐾) → 퐻푛(퐿).

In other words, 푓푛 does exactly what you would expect, and it simply forgets simplices that are “crushed down”.

Proof. Easy verification. See online notes for details.

Example. Retraction of a standard 2-simplex 퐾 to standard 1-simplex 퐿.

푓 푔 Remark. The map is functorial, i.e. given simplicial maps 퐾 −→퐿 −→푀, (푔 ∘ 푓) = 푔 ∘ 푓 ( ) = ∗ ∗ ∗. In addition id퐾 ∗ id퐻푛(퐾).

Definition (chain homotopy). Let 푓•, 푔• ∶ 퐶• → 퐷• be chain maps. A chain homotopy ℎ• between 푓• and 푔• is a sequence of homomorphisms ℎ푛 ∶ 퐶푛 → 퐷푛+1 such that

푔푛 − 푓푛 = 휕푛+1 ∘ ℎ푛 + ℎ푛−1 ∘ 휕푛

푛 푓 ≃ 푔 푓 ≃ 푔 for all . Write • • or • ℎ• •.

Lemma 5.5. If 푓• ≃ 푔• ∶ 퐶• → 퐷• then

푓∗ = 푔∗ ∶ 퐻푛(퐶•) → 퐻푛(퐷•)

for all 푛.

33 5 Homology

Proof. Consider [푐] ∈ 퐻푛(퐶•) so 푐 ∈ 푍푛(퐶•) = ker 휕푛. Then

푔푛(푐) − 푓푛(푐) = 휕푛+1 ∘ ℎ푛(푐) + ℎ⏟⏟⏟⏟⏟푛−1 ∘ 휕푛(푐) ∈ 퐵푛(퐷•) =0 so [푔푛(푐)] = [푓푛(푐)] so 푓∗ = 푔∗ as claimed. Example. Continuation of the previous example.

Definition (cone). A simplicial complex 퐾 is a cone if there is a vertex 푥0 such that for every 휏 ∈ 퐾 there exists 휎 ∈ 퐾 such that 푥0 ∈ 휎 and 휏 ≤ 휎.

Lemma 5.6. If 퐾 is a cone then it has the same homology as a point, i.e.

푛 = 0 퐻 (퐾) = {Z 푛 0 푛 > 0

Proof. Let 푥0 be a point as in the definition of a cone. Consider

푖 ∶ {⟨푥0⟩} → 퐾

푟 ∶ 퐾 → {⟨푥0⟩} 푟∘푖 = 푟 ∘푖 = the obvious inclusion and retraction. Clearly id{⟨푥0⟩} so ∗ ∗ id퐻푛({⟨푥0⟩}) 푛 푖 ∘ 푟 ≃ 푖 ∘ 푟 = for all . Thus left to show • • id퐶•(퐾) as if so then ∗ ∗ id퐻푛(퐾) for all 푛 so 푟∗ is an isomorphism and the result follows. We write down the following chain homotopy

ℎ푛 ∶ 퐶푛(퐾) → 퐶푛+1(퐾)

⟨푥0, 푣0, … , 푣푛⟩ 푥0 ∉ 휎 ⟨푣0, … , 푣푛⟩ ↦ { 0 푥0 ∈ 휎 Now check directly that ( −푖 ∘ 푟 )(휎) = (휕 ∘ ℎ + ℎ ∘ 휕 )(휎) id퐶푛(퐾) 푛 푛 푛+1 푛 푛−1 푛 for all 휎 ∈ 퐾. There are 4 cases depending on if 푥0 ∈ 휎 and if 푛 = 0. We’ll do the case 푥0 ∈ 휎, 푛 ≠ 0. The others are similar but easier. Let 휎 = ⟨푣0, … , 푣푛⟩ and suppose 푥0 = 푣푗. Now (휕 ∘ ℎ + ℎ ∘ 휕)(휎) = ℎ ∘ 휕(휎) 푛 푖 = ℎ(∑(−1) ⟨푣, … ,푖 ̂푣 , … , 푣푛⟩) 푖=0 푗 = (−1) ⟨푥0, 푣0, … , 푣푗−1, 푣푗+1, … , 푣푛⟩ 푗 푗 = (−1) (−1) ⟨푣0, … , 푣푗−1, 푥0, 푣푗+1, … , 푣푛⟩ = 휎 = ( −푖 ∘ 푟 )(휎) id퐶푛(퐾) 푛 푛

34 5 Homology

5.3 Homology of the simplex and the sphere

Example. Let 퐾 = {휏 ≤ 휎푛} where 휎푛 is the standard 푛-simplex. By consid- ering any vertex, 퐾 is obviously a cone so by the lemma

푛 = 0 퐻 (퐾) ≅ {Z 푛 0 푛 ≥ 1

Example. Let 퐿 = 휕휎푛 ⊆ 퐾 be the standard (푛 − 1)-sphere where 푛 ≥ 2. In other words 퐿 = 퐾 − {휎푛}.

0 = 퐶푛+1(퐿) 퐶푛(퐿) = 0 퐶푛−1(퐿) 퐶푛−2(퐿) ⋯ 퐶1(퐿) 퐶0(퐿) 0

0 = 퐶푛+1(퐾) 퐶푛(퐾) = ⟨휎푛⟩ 퐶푛−1(퐾) 퐶푛−2(퐾) ⋯ 퐶1(퐾) 퐶0(퐾) 0

So for 푘 ≤ 푛−2, 퐻푘(퐿) = 퐻푘(퐾). The only interesting case is 푘 = 푛−1. Because 퐻푛−1(퐾) ≅ 0, 푍푛−1(퐾) = 퐵푛−1(퐾) and similarly 푍푛(퐾) = 퐵푛(퐾) = 0 so 휕푛 is injective. Because 퐵푛−1(퐿) ≅ 0,

퐻푛−1(퐿) = 푍푛−1(퐿)/퐵푛−1(퐿) ≅ 푍푛−1(퐿)

= 푍푛−1(퐾)

= 퐵푛−1(퐾)

= im 휕푛

≅ 퐶푛(퐾) ≅ Z so 푘 = 0, 푛 − 1 퐻 (퐿) ≅ {Z 푘 0 푘 ≥ 2

Intuitively the homology groups detect 푛 − 1-dimensional holes (compare 푛 to 퐷 ). It also gives something that 휋1 fails to detect and gives us a way to differentiate 푆푛−1 for different 푛. (?)

5.4 Continuous maps and homotopies Question: if 휙 ∶ |퐾| → |퐿| is continuous, does it induce some kind of map 휙∗ ∶ 퐻푛(퐾) → 퐻푛(퐿)? The obvious idea to take simplicial approximation (푟) (푟) 푓 ∶ 퐾 → 퐿 of 휙 and set 휙∗ = 푓∗ ∶ 퐻푛(퐾 ) → 퐻푛(퐿). This brings two immediate problems: in general this 푟 is not 1, and moreover 휙∗ may depend on the choice of 푓.

Definition (continguous). Two simplicial maps 푓, 푔 ∶ 퐾 → 퐿 are contingu- ous if for every 휎 ∈ 퐾 there exists 휏 ∈ 퐿 such that 푓(휎), 푔(휎) ≤ 휏.

Informally this is the homotopy of simplicial maps. Remark. Look back at the proof of lemma 4.25, we proved that if 푓 is a simplicial approxiamtion to 휙 and if 푥 ∈ 휎, 휙(푥) ∈ ̊휏 then 푓(휎) ≤ 휏. Therefore if 푓, 푔 are both simplicial approximation to 휙 then they are contiguous.

35 5 Homology

Lemma 5.7. If 푓, 푔 ∶ 퐾 → 퐿 are contiguous then

푓∗ = 푔∗ ∶ 퐻푛(퐾) → 퐻푛(퐿)

for all 푛.

Proof. Need to exhibit a chain homotopy between 푓• and 푔•. Fix a total ordering < on th vertices of 퐾. Now for each simplex 휎 ∈ 퐾 we can write it in a unqiue way 휎 = ⟨푣0, … , 푣푛⟩ such that

푣0 < 푣1 < ⋯ < 푣푛. Now define a chain homotopy

ℎ푛 ∶ 퐶푛(퐾) → 퐶푛+1(퐿) 푛 푗 ⟨푣0, … , 푣푛⟩ ↦ ∑(−1) ⟨푓(푣0), … , 푓(푣푗), 푔(푣푗), … , 푔(푣푛)⟩ 푗=0

⟨푓(푣0), … , 푓(푣푗), 푔(푣푗), … , 푔(푣푛)⟩ = 0 if it is not an (푛 + 1)-dimensional simplex. We can now check directly that this defines a chain homotopy. See online notes for details.

Lemma 5.8. Let 퐾 be a simplicial complex. A simplicial map 푠 ∶ 퐾′ → 퐾 is a simplicial approximation to the identity if and only if 푠( ̂휎) is a vertex of 휎 for all 휎 ∈ 퐾. Futhermore such an 푠 exists. Proof. In the setting, the definition of simplicial approximation tells us that

id|퐾|(St퐾′ ( ̂휎)) = ̊휎⊆ St퐾(푠( ̂휎)), which hapens if and only if 휎( ̂휎) is a vertex of 휎. For all 휎 ∈ 퐾, choose any vertex of 휎 and assign 푠( ̂휎) to it. A simplex of 퐾 is the form ⟨ ̂휎0, … , ̂휎푛⟩ such that 휎0 ≤ 휎1 ≤ ⋯ ≤ 휎푛 so every 푠( ̂휎푖) is a vertex of 휎푛. Therefore ⟨푠( ̂휎0), … , 푠( ̂휎푛)⟩ is a face of 휎푛, so 푠 is a simplicial map. ′ The choice of 푠 induces a canonical homomorphism 푠∗ ∶ 퐻푛(퐾 ) → 퐻푛(퐾).

′ Proposition 5.9. 푠∗ ∶ 퐻푛(퐾 ) → 퐻푛(퐾) is an isomorphism for all 푛. Proof. Postponed until Mayer-Vietoris sequence.

Definition. Let 훼 ∶ |퐾| → 푋 be a triangulation we define

퐻푛(푋) = 퐻푛(퐾). Let 휙 ∶ 푋 → 푌 be continuous and 훼 ∶ |퐾| → 푋, 훽 ∶ |퐾| → 푌 be triangu- lations. Let 푓 ∶ 퐾(푟) → 퐿 be a simplicial approximation to 훽−1 ∘ 휙 ∘ 훼. Using (푟) simplicial approximation to the identity, we identify 퐻푛(퐾 ) = 퐻푛(퐾) for all 푟. Now set (푟) 휙∗ = 푓∗ ∶ 퐻푛(푋) = 퐻푛(퐾) = 퐻푛(퐾 ) → 퐻푛(퐿) = 퐻푛(푌 ).

By results we have proven, 휙∗ is independent of the choice of simplicial approx- imation. However, we want something stronger: we want homology to be a homotopy invariant.

36 5 Homology

Theorem 5.10. If 푋, 푌 are triangulable spaces and 휙, 휓 ∶ 푋 → 푌 are homotopic. Then 휙∗ = 휓∗.

Non-examinable. Sketch of proof. Let 훼 ∶ |퐾| → 푋, 훽 ∶ |퐿| → 푌. By hypothesis

훽−1 ∘ 휙 ∘ 훼 ≃ 훽−1 ∘ 휓 ∘ 훼 ∶ |퐾| → |퐿| are homotopic. Let Ψ ∶ |퐾| × 퐼 → |퐿| be such a homotopy. By example sheet 3 Q9 |퐾| × 퐼 ≅ |푀| for some simplicial complex 푀, such that the “top” and “bottom” 퐾0, 퐾1 ≅ 퐾 and embeds in 푀 via 푖∶퐾→푀,푗∶퐾→푀, and for all 휎 ∈ 퐾 there exists 푀휎 ⊆ 푀 such that |푀휎| ≅ 휎 × 퐼. Note that

|휕푀휎| = (휎 × {0}) ∪ (휎 × {1}) ∪ 푀휕휎.

Define a chain homotopy

(푟) (푟) ℎ푛 ∶ 퐶푛(퐾 ) → 퐶푛+1(푀 ) 휎 ↦ ∑ ⋯

Interpreting the equation about 휕푀휎 as a statement about oriented simplices,

휕 ∘ ℎ(휎) = 푗(휎) − 푖(휎) − ℎ ∘ 휕(휎) so 푗(휎) − 푖(휎) = 휕 ∘ ℎ(휎) + ℎ ∘ 휕(휎). Note that 퐹 ∘ 푖 is a simplicial approximation to 휙 = Φ ∘ 푖 and 퐹 ∘ 푗 is a simplicial approximation to 휏 = Φ ∘ 푗. Thus

퐹 ∘ 푗(휎) − 퐹 ∘ 푖(휎) = 퐹 ∘ 휕 ∘ ℎ(휎) + 퐹 ∘ ℎ ∘ 휕(휎) = 휕 ∘ (퐹 ∘ ℎ)(휎) + (퐹 ∘ ℎ) ∘ 휕(휎) so 퐹 ∘ ℎ is a homotopy between 퐹 ∘ 푖 and 퐹 ∘ 푗. Thus

휙∗ = (퐹 ∘ 푖)∗ = (퐹 ∘ 푗)∗ = 휓∗.

휙 휓 Lemma 5.11. If 푋, 푌 , 푍 are triangulable and 푋 −→푌 −→푍 then

(휓 ∘ 휙)∗ = 휓∗ ∘ 휙∗.

( ) = Also id푋 ∗ id퐻푛(푋).

Proof. Omitted.

Corollary 5.12. If 푋, 푌 are triangulable and 휙 ∶ 푋 → 푌 is a homotopy equivalence then 휙∗ ∶ 퐻푛(푋) → 퐻푛(푌 ) is an isomorphism for all 푛.

In other words, homology is a homotopy invariance.

37 6 Homology calculations

6 Homology calculations

6.1 Homology of spheres and applications Example. 푆푛−1 ≅ |퐿| where 퐿 is the standard simplicial (푛−1)-sphere. There- fore 푛−1 Z 푘 = 0, 푛 − 1 퐻푘(푆 ) ≅ 퐻푘(퐿) = { 0 otherwise Two implications: first since

푛−1 퐻푛−1(푆 ) ≅ Z ≇ 0 ≅ 퐻푛−1(∗)

푆푛−1 is not contractible. Secondly since

푛−1 푚−1 퐻푛−1(푆 ) ≅ Z ≇ 0 ≅ 퐻푛−1(푆 ) for 푚 ≠ 푛 we see 푆푛−1 ≄ 푆푚−1 unless 푛 = 푚.

Theorem 6.1 (invariance of domain). If R푚 ≅ R푛 then 푚 = 푛.

Proof. Suppose 휙 ∶ R푚 → R푛 is a homeomorphism. wlog 휙(0) = 0. This induces a homeomorphism R푚 \ {0} ≅ R푛 \ {0}. But they are homotopy equivalent to 푆푚−1 and 푆푛−1 respectively. 푚 = 푛.

Theorem 6.2 (Brouwer fixed point theorem). Let 퐷푛 be the closed 푛- dimensional disk. Then any continuous map 휙 ∶ 퐷푛 → 퐷푛 has a fixed point.

Proof. Identical to the 2 dimensional case, substituting 퐻푛−1 for 휋1.

6.2 Mayer-Vietoris theorem

Definition ((short) exact sequence). A sequence of homomorphism of abelian groups

푓푖+1 푓푖 ⋯ 퐴푖+1 퐴푖 퐴푖−1 ⋯

is exact at 퐴푖 if ker 푓푖 = im 푓푖+1. The sequence is exact if it is exact at every 퐴푖. A short exact sequence is one of the form

0 퐴 퐵 퐶 0

Example.

푓 1. 퐴 −→퐵 → 0 is exact at 퐵 if and only if 푓 is surjective.

푓 2. 0 → 퐴 −→퐵 is exact at 퐴 if and only if 푓 is injective.

38 6 Homology calculations

3. A very short exact sequence

푓 0 퐴 퐵 0

is an isomorphism 푓 ∶ 퐴 → 퐵.

Theorem 6.3 (Mayer-Vietoris). Let 퐾 = 퐿 ∪ 푀 with 푁 = 퐿 ∩ 푀 be simplicial complexes. Consider the inclusion maps

푁 푖 퐿

푗 ℓ 푀 푚 퐾

then there exists 훿∗ ∶ 퐻푛(퐾) → 퐻푛−1(푁) making this sequence exact.

⋯ 퐻푛+2(퐾)

훿∗ 푖∗⊕푗∗ ℓ∗−푚∗ 퐻푛+1(푁) 퐻푛+1(퐿) ⊕ 퐻푛+1(푀) 퐻푛+1(퐾)

훿∗ 푖∗⊕푗∗ ℓ∗−푚∗ 퐻푛(푁) 퐻푛(퐿) ⊕ 퐻푛(푀) 퐻푛(퐾)

훿∗ 푖∗⊕푗∗ ℓ∗−푚∗ 퐻푛−1(푁) 퐻푛−1(퐿) ⊕ 퐻푛−1(푀) 퐻푛−1(퐾)

훿∗ 퐻푛−1(푁) …

The core of the theorem is a result in homological algebra.

Definition (exact chain map). A sequence of chain maps

푓• 푔• 퐴• 퐵• 퐶•

is exact at 퐵• if 푓푛 푔푛 퐴푛 퐵푛 퐶푛

is exact at 퐵푛 for all 푛 ∈ Z.

Lemma 6.4 (snake lemma). Let

푓• 푔• 0 퐴• 퐵• 퐶• 0

be a short exact sequence of chain complexes. For any 푛 ∈ Z there is a

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homomorphism 훿∗ ∶ 퐻푛+1(퐶•) → 퐻푛(퐴•) such that

푔∗ ⋯ 퐻푛+1(퐶•)

훿∗ 푓∗ 푔∗ 퐻푛(퐴•) 퐻푛(퐵•) 퐻푛(퐶•)

훿∗ 푓∗ 퐻푛−1(퐴•) ⋯

Proof. Consider the massive commutative diagram

⋮ ⋮ ⋮

푓푛+1 푔푛+1 0 퐴푛+1 퐵푛+1 퐶푛+1 0

휕푛+1 휕푛+1 휕푛+1

푓푛 푔푛 0 퐴푛 퐵푛 퐶푛 0

휕푛 휕푛 휕푛

푓푛−1 푔푛−1 0 퐴푛−1 퐵푛−1 퐶푛−1 0

휕푛−1 휕푛−1 휕푛+1 ⋮ ⋮ ⋮

Let’s construct the map

훿∗ ∶ 퐻푛+1(퐶•) → 퐻푛(퐴•) [푥] ↦ ? with 푥 ∈ 푍푛+1(퐶•). Since 푔푛+1 is surjective, 푥 = 푔푛+1(푦) for some 푦 ∈ 퐵푛+1. Consider 휕푛+1(푦), by commutativity

푔푛 ∘ 휕푛+1(푦) = 휕푛+1 ∘ 푔푛(푦) = 휕푛+1(푥) = 0 as 푥 ∈ 푍푛+1(퐶•). Thus 휕푛+1(푦) ∈ ker 푔푛 = im 푓푛. Thus exists 푧 ∈ 퐴푛 such that 푓푛(푧) = 휕푛+1(푦). We would like to check that 푧 ∈ 푍푛(퐴•). Consider 휕푛(푧),

푓푛−1 ∘ 휕푛(푧) = 휕푛 ∘ 푓푛(푧) = 휕푛 ∘ 휕푛+1(푦) = 0.

But 푓푛−1 is injective so 휕푛(푧) = 0, and thus 푧 ∈ 푍푛(퐴•). Thus let 훿∗([푥]) = [푧]. We have to check this is well-defined and 훿∗ is a homomorphism. Finally we have to check exactness at 퐻푛(퐴•), 퐻푛(퐵•), 퐻푛(퐶•). This is just a tedious exercise in diagram chasing. Proof of Mayer-Vietoris. By the snake lemma it suffices to check that the fol- lowing is an exact sequence of chain complex.

푖•⊕푗• ℓ•−푚• 0 퐶•(푁) 퐶•(퐿) ⊕ 퐶•(푀) 퐶•(퐾) 0.

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• exactness at 퐶푛(푁): 푖푛 induces 퐶푛(푁) as a direct summand in 퐶푛(퐿) and similar for 푗푛. Thus 푖푛 ⊕ 푗푛 is injective.

• exactness at 퐶푛(퐾): consider 푐 ∈ 퐶푛(퐾), have 푐 = 푐퐿 + 푐푀 where 푐퐿, 푐푀 are “supported” in 퐿 and 푀 respectively. To make it precise, this means that they are respective images of ℓ푛 and 푚푛, i.e. there exists 푏퐿 ∈ 퐶푛(퐿), 푏푀 ∈ 퐶푛(푀) such that 푐퐿 = ℓ푛(푏퐿), 푐푀 = 푚푛(푏푀). Thus

푐 = ℓ푛(푏퐿) − 푚푛(−푏푀) = (ℓ푛 − 푚푛)(푏퐿, −푏푀).

• exactness at 퐶푛(퐿) ⊕ 퐶푛(푀): (푏퐿, 푏푀) ∈ ker(ℓ푛 − 푚푛) if and only if each simplex 휎 that appears in 푏퐿 also appears in 푏푀 with the same coefficients, if and only if (푏퐿, 푏푀) ∈ im(푖푛 ⊕ 푗푛).

Lemma 6.5 (five lemma). Suppose the following diagram is commutative and the rows are exact:

퐴 퐵 퐶 퐷 퐸

훼 훽 훾 훿 휀 퐴′ 퐵′ 퐶′ 퐷′ 퐸′

if 훼, 훽, 훿, 휀 are isomorphisms then so is 훾.

Proof. Example sheet 4. Recall that we claimed before given a barycentric subdivision 퐾′ of a sim- ′ plicial complex 퐾, the induced map 푠∗ ∶ 퐻푛(퐾 ) → 퐻푛(퐾) is an isomorphism. Proof. Induction on the number of simplices of 퐾. If 퐾 = {∗} then 퐾 = 퐾′ so obviously true. For inductive step, choose 휎 ∈ 퐾 with maximal dimension. Let

퐿 = 퐾 − {휎} 푀 = {휏 ≤ 휎 ∶ 휏 ∈ 퐾} 푁 = 퐿 ∩ 푀 = 휕휎

′ ′ 퐿, 푁 have fewer simplices than 퐾. Because 푀, 푀 are both cones, 푠∗ ∶ 퐻푛(푀 ) → ′ 퐻푛(푀) is an isomorphism. Look at Mayer-Vietoris for 퐾 = 퐿 ∪푁 푀, 퐾 = ′ ′ 퐿 ∪푁′ 푀 .

′ ′ ′ ′ ′ ′ ′ 퐻푛+1(푁 ) 퐻푛+1(퐿 ) ⊕ 퐻푛+1(푀 ) 퐻푛+1(퐾 ) 퐻푛(푁 ) 퐻푛(퐿 ) ⊕ 퐻푛(푀 )

푠∗ 푠∗⊕푠∗ 푠∗ 푠∗ 푠∗⊕푠∗

퐻푛+1(푁) 퐻푛+1(퐿) ⊕ 퐻푛+1(푀) 퐻푛+1(퐾) 퐻푛(푁) 퐻푛(퐿) ⊕ 퐻푛(푀)

so the five lemma finishes the job.

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6.3 Homology of compact surfaces Recall that classification of compact surfaces says there are two classes ofsur- faces there are two classes: Σ푔 and 푆푔. We are going to use Mayer-Vietoris to compute their homologies. Before that we have to know the homology of Γ푟. Γ = ⋁푟 푆1 Γ = ∗, Γ ≅ 푆1 Example. 푟 푖=1 . 0 1 and their homologies are known. Use standard simplicial 1-sphere. There is a slight issue here that the way we set up simplicial complex makes it difficult to glue things. Instead we are going to use abstract simplicial complex. 1 Let 퐾 be such that |퐾| ≅ Γ푟, 퐿, 푀 ⊆ 퐾 be such that |퐿| ≅ Γ푟−1, |푀| ≅ 푆 , and 퐾 = 퐿 ∪푁 푀 where 푁 = {푣0}. By Mayer-Vietoris,

퐻1(푁) 퐻1(퐿) ⊕ 퐻1(푀) 퐻1(퐾)

훿∗ 퐻0(푁) 퐻0(퐿) ⊕ 퐻0(푀) 퐻0(퐾)

0 which is

0 퐻1(Γ푟−1) ⊕ Z 퐻1(Γ푟)

훿∗ Z Z ⊕ Z Z

0 and unfortunately we have to get out hands dirty to understand 훿∗. The map 퐻0(푁) → 퐻0(퐿)⊕퐻0(푀) sends generator to generators so is injective and thus by exactness at 퐻1(퐾), im 훿∗ = 0. Thus we have a very short exact sequence

훿∗ 0 퐻1(Γ푟−1) ⊕ Z 퐻1(Γ푟) 0 so 퐻1(Γ푟) ≅ Γ1(Γ푟−1) ⊕ Z. Thus we conclude that

⎧ 푛 = 0 {Z 퐻 (Γ ) = 푟 푛 푟 ⎨Z 푛 = 1 { ⎩0 otherwise

Note that 퐻1(Γ푟) ≅ ⟨[훼1]⟩⊕⋯⊕⟨[훼푟]⟩ where [훾푖] is a generator for 퐻1(푖th circle). Remark. Note that this proof does not assume anything other path-connectedness. Thus 훿∗ ∶ 퐻1(퐾) → 퐻0(푁) is always zero as long as the intersection 푁 is con- nected. Then we don’t have to worry about the last row of Mayer-Vietoris. Σ = Γ ∪ 퐷2 Example. 푔 2푔 휌푔 where

−1 −1 −1 −1 휌푔(1) = 훼1훽1훼1 훽1 ⋯ 훼푔훽푔훼푔 훽푔 .

42 6 Homology calculations

Just as mapping cylinder, to compute the homology it’s convenient to introduce a new space ∗ 1 Σ푔 = Γ2푔 ∪2푔 (푆 × [0, 1]) 1 2 with (푥, 0) ∼ 휌푔(푥) for all 푥 ∈ 푆 . This is just Σ푔 with 퐷 removed. Note that ∗ Σ푔 deformation retracts to Γ2푔. ∗ A bit of technical work: we can triangulate Σ푔 by triangulating Γ2푔 then 1 taking a simplicial approximation to 휌푔 and triangulating 푆 × 퐼 in a way com- pactible with that. Next note that ∗ 2 Σ푔 = Σ푔 ∪푖 퐷 1 ∗ 2 1 where 푖 ∶ 푆 → Σ푔 is the map identifying 휕퐷 with 푆 × 퐼. Choose a triangula- tion of 퐷2 compatible with the induced triangulation of its boundary. Back to the actual computation.

1 ∗ 2 퐻2(푆 ) 퐻2(Σ푔) ⊕ 퐻2(퐷 ) 퐻2(Σ푔)

1 ∗ 2 퐻1(푆 ) 퐻1(Σ푔) ⊕ 퐻1(퐷 ) 퐻1(Σ푔)

0 note that by the previous remark we don’t have to write down the last row. Fill in information we already knew,

푖∗ 2푔 0 0 퐻2(Σ푔) Z Z 퐻1(Σ푔) 0

To figure out the two unknown groups we need to understand 푖∗, which induced by 휌푔. But since homology groups are abelian, we have

푖∗(1) = (훾푔)∗(1) = [훼1] + [훽1] − [훼1] − [훽1] + ⋯ + [훼푔] + [훽푔] − [훼푔] − [훽푗] = 0 So we split that into two exact sequences

2푔 0 퐻2(Σ푔) Z 0 0 Z 퐻2(Σ푔) 0 so ⎧ 푛 = 0, 2 {Z 퐻 (Σ ) ≅ 2푔 푛 푔 ⎨Z 푛 = 1 { ⎩0 otherwise which in particular implies that they are pairwise non-homotopy equivalent. 푆 = Γ ∪ 퐷2 Example. 푔 푔+1 휎푔 where 2 2 2 휎푔(1) = 훼0훼1 ⋯ 훼푔. By almost the same argument, we obtain a Mayer-Vietoris sequence

훿∗ 푖∗ 푔+1 0 퐻2(푆푔) Z Z 퐻1(푆푔) 0 where this times 푖∗ is induced by 휎푔, and

푖∗ = (훾푔)∗(1) = 2[훼0] + 2[훼1] + ⋯ + 2[훼푔] 푔 which is injective so 훿∗ = 0. Thus 퐻2(푆푔) = 0, 퐻1(푆푔) ≅ Z ⊕ (Z/2Z).

43 6 Homology calculations

6.4 Rational homology and Euler characteristic Basically homology with coefficients in Q.

Definition (rational chain). Let 퐾 be a simplicial complex. The vector space of rational 푛-chain 퐶푛(퐾; Q) is defined as the vector space over Q with basis the 푛-simplices of 퐾. ∑ 휆 휎 휆 ∈ 휎 A typical element thus has the form 푖 푖 푖 where 푖 Q and 푖’s are 푛-simplices. We can define the boundary map 휕푛 ∶ 퐶푛(퐾; Q) → 퐶푛(퐾; Q) as before and the condition 휕 ∘ 휕 = 0 is satisfied so we have another homology theory.

Definition (rational homology). Let 퐾 be a simplicial complex. We define

푍푛(퐾; Q) = ker 휕푛

퐵푛(퐾; Q) = im 휕푛+1

퐻푛(퐾; Q) = 푍푛(퐾; Q)/퐵푛(퐾; Q)

If 훼 ∶ |퐾| → 푋 is a triangulation then define

퐻푛(푋; Q) = 퐻푛(퐾; Q). Rational homology encodes slightly less data but has the advantage of easier to compute (as Q is a field). More specifically, it simply forgets the torsion elements:

푘 Lemma 6.6. Let 퐾 be a simplicial complex. If 퐻푛(퐾) ≅ Z ⊕ 퐹 where 퐹 is a finite group then 푘 퐻푛(퐾; Q) ≅ Q . Proof. An exercise in commutative algebra. See online notes. Example. For all 푛, 2 퐻푛(R푃 ; Q) ≅ 퐻푛(∗; Q).

Definition (Euler characteristic). Let 푋 be a triangulable space and 훼 ∶ |퐾| → 푋 a triangulation. Then the Euler characteristic of 푋 is

휒(푋) = 휒(퐾) = ∑(−1)푛 퐻 (퐾; ). dimQ 푛 Q 푛∈Z Note that this is obviously a topological invariant. Given the way rational homology is defined, the nice thing about 휒(퐾) is that it is really easy to compute.

Lemma 6.7. Let 퐾 be a simplicial complex. Then

휒(퐾) = ∑(−1)푛#{푛-simplicies in 퐾}. 푛∈Z In particular if 퐾 is 2-dimensional, with 푉 , 퐸, 퐹 the number of 0, 1 and

44 6 Homology calculations

2-simplices then 휒(퐾) = 푉 − 퐸 + 퐹 . Proof. The number of 푛-simplex is not a natural algebraic object so instead we 퐶 (퐾; ) use dimQ 푛 Q . Next, recall that we are working with vector spaces so we can apply rank-nullity theorem to

푍푛(퐾; Q) ↠ 퐻푛(퐾; Q)

휕푛 ∶ 퐶푛 → 퐵푛−1 to get

dim 푍푛 = dim 퐻푛 + dim 퐵푛

dim 퐶푛 = dim 퐵푛−1 + dim 푍푛

Now

∑(−1)푛 퐶 = ∑(−1)푛 퐵 + ∑(−1)푛 푍 dimQ 푛 dim 푛−1 dim 푛 푛∈Z 푛∈Z 푛∈Z 푛 푛 = − ∑(−1) dim 퐵푛 + ∑(−1) dim 푍푛 푛∈Z 푛∈Z 푛 = ∑(−1) (dim 푍푛 − dim 퐵푛) 푛∈Z 푛 = ∑(−1) dim 퐻푛 푛∈Z = 휒(퐾)

Note.

휒(Σ푔) = 2 − 2푔

휒(푆푔) = 1 − 푔

6.5 Lefschetz fixed-point theorem

Definition (Lefschetz number). Let 휙 ∶ 푋 → 푋 be a continuous map of triangulable space 푋. The Lefschetz number of 휙 is

푛 퐿(휙) = ∑(−1) tr(휙∗ ∶ 퐻푛(푋; Q) → 퐻푛(푋; Q)). 푛∈Z

Note. 퐿(id푋) = 휒(푋) so Lefschetz number generalises Euler characteristic.

Lemma 6.8. If 푓 ∶ 퐾 → 퐾 is a simplicial map then

푛 퐿(|푓|) = ∑(−1) tr(푓∗ ∶ 퐶푛(퐾; Q) → 퐶푛(퐾; Q)). 푛∈Z

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Proof. Consider the following commutative diagrams of linear maps with exact rows: 0 퐴 퐵 퐶 0 훼 훽 훾 0 퐴′ 퐵′ 퐶′ 0 By linear algebra we can find bases for 퐵, 퐵′ such that the matrix for 훽 have has the form 훾 ∗ ( ) 0 훼 so in particular tr 훽 = tr 훾 + tr 훼. What is left is an application of rank-nullity theorem, similar to the proof of Euler characteristic.

Theorem 6.9 (Lefschetz fixed point theorem). Let 푋 be a triangulable space and 휙 ∶ 푋 → 푋 a continuous map. If 퐿(휙) ≠ 0 then 휙 has a fixed point.

Non-examinable, sketch. Suppose 휙 has no fixed point. Since 푋 is compact, exists 훿 > 0 such that for all 푥 ∈ 푋, ‖푥 − 휙(푥)‖ > 훿. Now choose a simplicial 훿 (푟) approximation 퐾 of 푋 with mesh(퐾) < 2 . Let 푓 ∶ 퐾 → 퐾 be a simplicial approximation to 휙. Note that if 푣 ∈ 휎 ∈ 휎 then 푓(푣) ∉ 휎. Let 휄푛 ∶ 퐶푛(퐾; Q) → 퐶푛(퐾; Q) be the map inducing canonical isomorphism of homology groups. For any 푛-simplex 휎 ∈ 퐾, 휄푛(휎) is supported on simplices contained in 휎. Then 푓푛 ∘ 휄푛 takes every simplex of 퐾 off itself, i.e. 푓푛 ∘ 휄푛(휎) does not contain 휎. Now

푛 퐿(휙) = ∑(−1) tr(푓푛 ∘ 휄푛 ∶ 퐶푛(퐾; Q) → 퐶푛(퐾; Q)) 푛∈Z = ∑(−1)푛 ⋅ 0 푛∈Z = 0

46 Index barycentre, 26 Lefschetz fixed point theorem, 46 barycentric subdivision, 26, 41 Lefschetz number, 45 boundary, 24, 30, 32 lift, 9 boundary homomorphism, 29 loop, 4 Brouwer fixed point theorem, 13, 38 Mayer-Vietoris theorem, 39 mesh, 26 cell, 21 chain, 29 neighbourhood deformation chain complex, 32 retract, 20 chain homotopy, 33 classification of compact surfaces, orientation, 29 23 cone, 34 path, 4 contiguous, 35 path-lifting lemma, 9 contractible, 3 polyhedron, 25 covering space, 8 presentation, 17 covering space isomorphism, 14 pushout, 17 covering transformation, 15 rational chain, 44 cycle, 30, 32 rational homology, 44 deck transformation, 15 realisation, 25 deformation retract, 4 retract, 4 degree, 10 Seifert-van Kampen, 20 dimension, 24 Seifert-van Kampen theorem, 19 Euler characteristic, 44 short exact sequence, 38, 39 exact sequence, 38 simplex, 24 simplicial approximation, 27 face, 24 simplicial approximation theorem, five lemma, 41 27 free group, 16 simplicial complex, 24 free product, 18 simplicial map, 25 amalgamated, 18 realisation, 25 fundamental group, 5 simply connected, 7 skeleton, 24 Galois correspondence, 14 snake lemma, 39 star, 27 homology, 32 surface, 22 homology group, 30 homotopy, 3 topological manifold, 22 homotopy equivalence, 3 triangulation, 25 homotopy lifting lemma, 10 homotopy of path, 4 universal cover, 12 invariance of domain, 38 wedge, 19

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