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.