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Fibre Bundles Fibre Bundles Andrew Kobin Spring 2017 / Spring 2018 Contents Contents Contents 0 Introduction 1 1 Fibre Bundles 2 1.1 Locally Trivial Fibre Bundles . .2 1.2 Principal Bundles . .5 1.3 Classifying Spaces and Universal Bundles . 10 1.4 Reduction of the Structure Group . 16 1.5 Gauge Groups . 19 2 Vector Bundles 22 2.1 Vector and Tangent Bundles . 22 2.2 Sections . 25 2.3 The Induced Bundle . 28 2.4 Bundle Operations . 30 3 Characteristic Classes 34 3.1 Metrics on Vector Bundles . 34 3.2 Stiefel-Whitney Classes . 35 3.3 The Universal Bundle . 44 3.4 Construction of Stiefel-Whitney Classes . 51 3.5 Oriented Bundles . 53 3.6 Chern Classes . 55 3.7 Pontrjagin Classes . 57 3.8 General Characteristic Classes . 58 4 K-Theory 70 4.1 K-Theory and Reduced K-Theory . 71 4.2 External Products in K-theory . 73 4.3 Bott Periodicity . 75 4.4 Characteristic Classes in K-Theory . 82 5 Chern-Weil Theory 85 5.1 Connections on a Vector Bundle . 86 5.2 Characteristic Classes from Curvature . 92 5.3 The Levi-Civita Connection . 100 6 Index Theory 103 6.1 Differential Operators . 103 6.2 Hodge Decomposition . 108 6.3 The Index Theorem . 110 6.4 Dolbeault Cohomology . 113 i 0 Introduction 0 Introduction This document follows two courses in fibre bundles taught respectively by Dr. Slava Krushkal in Spring 2017 and Dr. Thomas Mark in Spring 2018 at the University of Virginia. The main concepts covered are: Locally trivial fibre bundles Principal bundles and classifying spaces Vector bundles and their important properties Characteristic classes, including Stiefel-Whitney, Chern and Pontryagin classes Orientability Euler class Elements of K-theory Connections and Chern-Weil theory The Atiyah-Singer index theorem. Standard references include: Milnor-Stasheff's Characteristic Classes, Hatcher's Vector Bun- dles and K-Theory and Husem¨oller's Fibre Bundles. 1 1 Fibre Bundles 1 Fibre Bundles 1.1 Locally Trivial Fibre Bundles Definition. A bundle is a triple (E; B; p) consisting of topological spaces E; B and a con- tinuous map between them, p : E ! B. We call E the total space of the bundle and B the base space. Definition. A morphism of bundles (E1;B1; p1) ! (E2;B2; p2) is a pair of maps f : ~ B1 ! B2; f : E1 ! E2, making the diagram f~ E1 E2 p1 p2 f B1 B2 −1 ~ −1 commute and preserving fibres, that is, for any x 2 B1 and y 2 p1 (x), f(y) 2 p2 (f(x)). Example 1.1.1. When f is the identity B ! B, we say the bundles and morphisms of bundles are over B. p Example 1.1.2. Let E −! B be a bundle and A ⊆ B be a subspace. Then the restriction −1 pjA : EjA := p (A) ! A is a bundle over A and the natural inclusion EjA ,! E induces a bundle morphism EjA E pjA p A B Example 1.1.3. Let F be any space. The trivial bundle over B with fibre F is the canonical projection p : B × F ! B. Definition. A locally trivial fibre bundle over B with fibre F , hereafter shortened to a p fibre bundle, is a bundle E −! B that is locally isomorphic to a trivial bundle with fibre F . That is, there exists a collection of open sets fUig together with bundle isomorphisms ∼ 'i : EjUi = Ui × F , called local trivializations. p Given a fibre bundle E −! B with locally trivial open sets fUig, there is a choice of −1 isomorphism 'i : EjUi ! Ui for each Ui. For each x 2 Ui, p (x) is homeomorphic to F via −1 'i, so when x 2 Ui \Uj, there are two different ways to identify p (x) with F . In particular, −1 'j ◦ 'i :(Ui \ Uj) × F ! (Ui \ Uj) × F is a homeomorphism and induces a homeomorphism F ! F . In other words, for each pair of open sets Ui;Uj, there is a function gij : Ui \ Uj −! Homeo(F ) 2 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles called a transition map. One can show that the gij are continuous. Note that the gij satisfy a few nice conditions, namely that gii = 1, gijgji = 1 and gijgjkgki = 1 pointwise on F . In fact, the data of a covering fUig and continuous transition maps fgijg satisfying these conditions are enough to determine a fibre bundle. Proposition 1.1.4. Let B and F be spaces and suppose there is an open cover fUig of B and continuous maps gij : Ui \ Uj ! Homeo(F ) satisfying: (a) For all x 2 Ui, gii(x) = 1, where 1 is the identity map F ! F . (b) For all x 2 Ui \ Uj, gij(x)gji(x) = 1. (c) For all x 2 Ui \ Uj \ Uk, gij(x)gjk(x)gki(x) = 1. Then there exists a fibre bundle over B with fibre F which is trivial over the Ui and has the gij as its transition maps. Moreover, such a bundle is unique up to isomorphism of bundles over B. Proof. (Sketch) Define the total space of the bundle by Y E := (Ui × F )= ∼ i where ∼ is the equivalence relation identifying (x; y) 2 Ui ×F and (x; gij(x)(y)) 2 Uj ×F for any x 2 Ui \ Uj and y 2 F . The bundle map p : E ! B is given by the natural projection on each disjoint piece, which is well-defined on all of E by (a) { (c) and is continuous since p the gij are continuous. It is straightforward to check that E −! B is a locally trivial bundle with fibre F having transition maps gij. p~ For uniqueness, suppose Ee −! B is another bundle with gij as its transition maps. For −1 each Ui, there is a fibre-preserving homeomorphismp ~ (Ui) ! Ui × F . Define a bundle map −1 Ee ! E by sending a pointx ~ 2 p (Ui) to the corresponding point in Ui × F ⊆ E. One now checks that this is a bundle isomorphism. p1 p2 Given two bundles E1 −! B and E2 −! B with fibre F and local trivializations f(Ui;'i)g and f(Vj; j)g, respectively, when can we tell if they are isomorphic? First, we may refine the covers fUig and fVjg by taking intersections until each bundle is trivial over both collections. Thus we may assume both bundles are trivial over fUig. Suppose f : E1 ! E2 is an isomorphism of bundles over B. Then each Ui corresponds to a local isomorphism of bundles −1 fi := i ◦ f ◦ 'i : Ui × F ! E1jUi ! E2jUi ! Ui × F: In fact, this gives us a necessary and sufficient condition for f to induce a bundle isomorphism. Lemma 1.1.5. The map f : E1 ! E2 is an isomorphism of bundles over B if and only if there exist local bundle isomorphisms fi : Ui × F ! Ui × F that are compatible with the −1 −1 transition maps 'j ◦ 'i and j ◦ i , i.e. the following diagram commutes: 3 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles fi (Ui \ Uj) × F (Ui \ Uj) × F −1 −1 'j ◦ 'i j ◦ i fj (Ui \ Uj) × F (Ui \ Uj) × F Example 1.1.6. Let M ⊆ RN be a smooth manifold and define the tangent bundle to M by N N TM = f(x; v) 2 R × R j x 2 M; v 2 TxMg: The projection p : TM ! M; (x; v) 7! x gives a fibre bundle structure on TM with fibre −1 ∼ n p (x) = TxM = R if M is an n-manifold. If f(Ui;'i)g is an atlas of coordinate charts for the manifold structure on M, then the derivatives of their transition maps, −1 gij : x 7−! d'i(x)('j ◦ 'i ); n define the transition maps gij : Ui \ Uj ! Homeo(R ) of the tangent bundle. Since these n derivatives are linear, the gij in fact have target GLn(R) ⊂ Homeo(R ). This reflects the fact that TM is a vector bundle (see Chapter 2). For example, let M = S2 be the 2-sphere. We may view S2 = CP 1, the complex projective line, which has a covering by two coordinate charts U0 = f[z0; z0] j z0 6= 0g and U1 = f[z0; z1] j z1 6= 0g z1 z0 with homeomorphisms '0 : U0 ! ; [z0; z1] 7! and '1 : U1 ! ; [z0; z1] 7! . Then C z0 C z1 −1 1 the transition maps 'j ◦ 'i : C ! C are both given by λ 7! λ . Thus the transition maps 2 1 g01; g10 for the tangent bundle TS are each given by λ 7! − λ2 , viewed as a 2×2 matrix with × × × real entries in GL2(R), or a scalar element of GL1(C) = C . Notice that g01 : C ! C is homotopic to a map S1 ! S1 of degree −2. It is no accident that S2 has Euler characteristic 2 { we will see this connection explicitly borne out by the Euler class (see Section 3.5). Let G be a topological group and F a space on which G acts by homeomorphisms, i.e. such that there is a continuous homomorphism ρ : G ! Homeo(F ). Definition. A fibre bundle with fibre F has structure group G if it has transition functions gij : Ui \ Uj ! G. Example 1.1.7. Fibre bundles with fibre F = Rn (resp. F = Cn) and structure group G = GLn(R) (resp. GLn(C)) are called real (resp. complex) vector bundles (see Chapter 2). Example 1.1.8. A bundle with fibre F = Sn and structure group G = O(n + 1) is called an (orthogonal) sphere bundle.
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