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Algebraic Topology Ensemble Algebraic topology ensemble Shintaro Fushida-Hardy 381B Sloan Hall, Stanford University, CA This document contains notes on algebraic topology - during my reading course with Ciprian Manolescu in Spring 2020, he noticed that I didn't really know any algebraic topology/homotopy theory beyond the first couple of chapters of Hatcher! After conclud- ing the reading course (on knot theory and 3-manifolds) we decided that I should learn some more algebraic topology, as it is fundamental to topology as a whole. In the first chapter, we introduce classifying spaces for vector bundles and principal G-bundles. In the second chapter we try to improve our classifications of vector bundles by introducing char- acteristic classes (and solve several exercises). We continue the trend of delving into vector bundles by studying K-theory in the third chapter, with an emphasis on the Bott period- icity theorem and classification of division algebras. This was my first experience with an extraordinary cohomology theory! Next we investigate another generalised (co)homology theory, namely cobordism theory. Finally we study some additional tools that can help us understand both K-theory and cobordism theory (and extraordinary cohomology theories in general): equivariant cohomology and the Atiyah-Hirzebruch spectral sequence. We give examples of calculations involving each of these. The primary sources are listed at the start of each chapter. Contents 1 Classifying spaces 3 1.1 Fibrations and fibre bundles . .3 1.2 Vector bundle and principal bundle basics . .7 1.3 Classifying spaces . 11 1.4 Examples of classifying spaces . 16 2 Characteristic classes 22 2.1 Characteristic classes in general . 22 2.2 The Stiefel-Whitney class . 25 2.3 The Chern class . 31 2.4 Applications of the Stiefel-Whitney and Chern classes . 33 2.5 The Euler class . 34 2.6 The Pontryagin class . 39 2.7 Exercises . 40 3 K-theory 50 3.1 Basic definitions . 50 3.2 K-theory as cohomology, and the Bott periodicity theorem . 55 3.3 Bott periodicity: applications and examples . 61 3.4 K-theory versus singular cohomology . 73 4 (Co)bordism theory 76 4.1 Unoriented bordism as a homology theory . 76 4.2 G-Bordism and the Pontryagin-Thom construction . 82 5 Equivariant cohomology 87 5.1 Equivariant cohomology fundamentals . 87 5.2 Leray spectral sequence . 92 5.3 Localisation theorem . 94 5.4 Equivariant K-theory . 95 1 6 Atiyah-Hirzebruch spectral sequence 101 6.1 AHSS in general . 101 6.2 AHSS applied to K-theory and cobordism theory . 102 2 Chapter 1 Classifying spaces Fibre bundles (and in particular vector bundles) appear throughout maths, so we want some way of classifying bundles over a given space X. Given a topological group G, we will construct the classifying space BG, and show that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with homotopy classes of maps X ! BG. In this chapter, we start by recalling general definitions for fibre bundles and fibrations. We then look at the special cases of vector bundles and principal bundles, before constructing classifying spaces and establishing the main result mentioned above. Finally we will study some examples. This chapter is primarily sourced from [Kot12, Mit06, Coh98]. 1.1 Fibrations and fibre bundles The most general notion we consider is a fibration. Most diagrams we consider will involve fibrations or cofibrations. Covering spaces and fibre bundles are also examples of fibrations. Definition 1.1.1. A fibration p : E ! B is a map satisfying the homotopy lifting property. That is, any homotopy f : A × I ! B lifts to fe : A × I ! E. As a consequence of this definition, the fibre of any two points in B are homotopy equivalent. Therefore we speak of the fibre F of a fibration, and write p F −!i E −! B where p is the fibration and i is an inclusion. Although not all maps are fibrations, we can approximate all maps by fibrations. Proposition 1.1.2. Let f : X ! Y be any map between topological spaces. There exists a fibration fe : Xe ! Y which approximates f in the sense that there is a homotopy equivalence h : X ! Xe such that the following diagram commutes: 3 Xe h fe f X Y: Proof outline. Define Xe := f(x; γ) 2 X × Y I such that γ(0) = f(x).g Here Y I is the space of all paths I = [0; 1] ! Y equipped with the compact-open topology. This is called the mapping path space of f. Define fe : Xe ! Y by fe :(x; γ) 7! γ(1): We claim without proof that this map is a fibration. Next we define h : X ! Xe by h : x 7! (x; γx); where γx is the constant path γx(t) = f(x). It is clear that the above diagram commutes, because (fe ◦ h)(x) = fe(x; γx) = γx(1) = f(x). Finally, to see that h is a homotopy equivalence, define g : Xe ! X by g :(x; γ) 7! x: One can show that h ◦ g and g ◦ h are homotopic to the identity. The mapping path space introduced in the above proof is a canonical choice for Xe, which also gives us a canonical choice of fibre. Definition 1.1.3. Let f : X ! Y be a map. We have the following commutative diagram: i Ff Ef h p fe f X Y: The map fe : Ef ! Y is a fibration, where Ef is the mapping path space defined by I Ef := f(x; γ) 2 X × Y : γ(0) = f(x)g: Ef is homotopic to X, so fe can be seen as a fibration that approximates f. Ff denotes the homotopy fibre of f, which is the pullback of the above diagram, and the fibre of fe. Explicitly, we have I Ff := f(x; γ) 2 X × Y : γ(0) = f(x); γ(1) = y0g; where y0 is a base point of Y . The map p is the projection onto the first coordinate, and i is the inclusion map. 4 The most important theorem concerning fibrations is that they extend to a long exact sequence. Recall that given a pointed space X, the loop space of X, denoted ΩX, is the 1 space of pointed maps from S into X equipped with the compact-open topology. Let F ! E ! B be a fibre sequence (i.e. p : E ! B is a fibration, with fibre F ). Replacing the fibre sequence with Fp ! Ep ! B, one can show that i : Fp ! Ep is itself a fibration, with fibre ΩB. To see why the fibre is ΩB, we simply write it out explicitly: −1 ∼ I i ((e0; γ)) = f(e0; γ): γ(0) = f(e0) = b0 = γ(1)g = fγ 2 B : γ(0) = γ(1) = b0g = ΩB: Given a map f : X ! Y , we also have a canonical map f 0 :ΩX ! ΩY defined by f 0 : γ ! f ◦ γ: Combining these constructions actually gives a long fibre sequence, as described in the following theorem. Theorem 1.1.4. Let F ! E ! B be a fibre sequence. Then there is a long fibre sequence ···! Ω2B ! Ω1F ! Ω1E ! Ω1B ! F ! E ! B: By a long fibre sequence, we mean that any two consecutive maps define a fibre sequence. Corollary 1.1.5. If F ! E ! B is a fibre sequence, there is an associated long exact sequence of homotopy ···! πn+1(B) ! πn(F ) ! πn(E) ! πn(B) ! πn−1(F ) !··· : Proof outline. We induce a long exact sequence 0 n+1 0 n 0 n 0 n ···! [S ; Ω B]0 ! [S ; Ω F ]0 ! [S ; Ω E]0 ! [S ; Ω B]0 !··· : Recall that [X; Y ] denotes the homotopy classes of maps from X to Y , and here we have used [X; Y ]0 to denote the classes of based maps. By Eckmann-Hilton duality, we have 0 n n 0 n [S ; Ω X]0 = [Σ S ;X]0 = [S ;X]0 = πn(X); where ΣY denotes the suspension of Y . The result follows. This is a key result which we wish to apply in many scenarios, an in particular in the context of bundles. It turns out that under mild conditions, bundles are indeed fibrations and we have a long exact sequence of homotopy groups. Definition 1.1.6. A fibre bundle is the data (π; E; B; F ), where π is a locally trivial surjection π : E ! B of topological spaces. E is called the total space, and B the base space, and F the fibre space. Locally trivial means that for any b 2 B there is a neighbourhood U of b so that π−1(U) ⊂ E is homeomorphic to F × U, and the following diagram commutes: 5 π−1(U) U × F π proj B: Definition 1.1.7. A morphism of fibre bundles F :(π; E; B) ! (π0;E0;B0) is a pair (fE; fB) such that the following diagram commutes: f E E E0 π π0 f B B B0: We often write π : E ! B, or even just E, to denote a fibre bundle. A morphism of fibre bundles is often denoted by f : E ! E0. Remark. The collection of fibre bundles forms a category, with morphisms as defined above. Fixing a base space gives a subcategory. Next we observe that for both fibrations and fibre bundles, we have pullbacks. Definition 1.1.8. Let π : E ! B be a fibre bundle, and f : A ! B continuous. The pullback bundle π0 : f ∗(E) ! A is defined to be the pullback of the diagram E ! B A. Explicitly, define ∗ f (E) = A ×B E = f(a; e) 2 A × E : f(a) = π(e)g; and consider the diagram proj f ∗(E) 2 E proj1 π f A B: Then the arrow on the left defines the pullback bundle.
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