The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector Bundles 3 1.2. Lie Groups and Principal Bundles 7 1.3. Clutching Functions and Structure Groups 15 2. Pull Backs and Bundle Algebra 21 2.1. Pull Backs 21 2.2. The tangent bundle of Projective Space 24 2.3. K - theory 25 2.4. Differential Forms 30 2.5. Connections and Curvature 33 2.6. The Levi - Civita Connection 39 Chapter 2. Classification of Bundles 45 1. The homotopy invariance of fiber bundles 45 2. Universal bundles and classifying spaces 50 3. Classifying Gauge Groups 60 4. Existence of universal bundles: the Milnor join construction and the simplicial classifying space 63 4.1. The join construction 63 4.2. Simplicial spaces and classifying spaces 66 5. Some Applications 72 5.1. Line bundles over projective spaces 73 5.2. Structures on bundles and homotopy liftings 74 5.3. Embedded bundles and K -theory 77 5.4. Representations and flat connections 78 Chapter 3. Characteristic Classes 81 1. Preliminaries 81 2. Chern Classes and Stiefel - Whitney Classes 85 iii iv CONTENTS 2.1. The Thom Isomorphism Theorem 88 2.2. The Gysin sequence 94 2.3. Proof of theorem 3.5 95 3. The product formula and the splitting principle 97 4. Applications 102 4.1. Characteristic classes of manifolds 102 4.2. Normal bundles and immersions 105 5. Pontrjagin Classes 108 5.1. Orientations and Complex Conjugates 109 5.2. Pontrjagin classes 111 5.3. Oriented characteristic classes 114 6. Connections, Curvature, and Characteristic Classes 115 Chapter 4. Homotopy Theory of Fibrations 125 1. Homotopy Groups 125 2. Fibrations 130 3. Obstruction Theory 135 4. Eilenberg - MacLane Spaces 140 4.1. Obstruction theory and the existence of Eilenberg - MacLane spaces 140 4.2. The Hopf - Whitney theorem and the classification theorem for Eilenberg - MacLane spaces 144 5. Spectral Sequences 152 5.1. The spectral sequence of a filtration 152 5.2. The Leray - Serre spectral sequence for a fibration 157 5.3. Applications I: The Hurewicz theorem 160 n ∗ 5.4. Applications II: H∗(ΩS ) and H (U(n)) 162 5.5. Applications III: Spin and SpinC structures 165 Bibliography 171 Introduction Fiber Bundles and more general fibrations are basic objects of study in many areas of mathe- matics. A fiber bundle with base space B and fiber F can be viewed as a parameterized family of objects, each “isomorphic” to F , where the family is parameterized by points in B. For example a vector bundle over a space B is a parameterized family of vector spaces Vx, one for each point x ∈ B. Given a Lie group G, a principal G - bundle over a space B can be viewed as a parameterized family of spaces Fx, each with a free, transitive action of G (so in particular each Fx is homeomorphic to G). A covering space is also an example of a fiber bundle where the fibers are discrete sets. Sheaves and “fibrations” are generalizations of the notion of fiber bundles and are fundamental objects in Algebraic Geometry and Algebraic Topology, respectively. Fiber bundles and fibrations encode topological and geometric information about the spaces over which they are defined. Here are but a few observations on their impact in mathematics. • A structure such as an orientation, a framing, an almost complex structure, a spin structure, and a Riemannian metric are all constructions on the the tangent bundle of a manifold. • The exact sequence in homotopy groups, and the Leray - Serre spectral sequence for ho- mology groups of a fibration have been basic tools in Algebraic Topology for nearly half a century. • Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. • K - theory, a type of classification of vector bundles over a topological space is at the same time an important homotopy invariant of the space, and a quantity for encoding index information about elliptic differential operators. • The Yang - Mills partial differential equations are defined on the space of connections on a principal bundle over a Riemannian two dimensional or four dimensional manifold. The properties of the solution space have had a great impact on our understanding of four dimensional Differential Topology in the last fifteen years. In these notes we will study basic topological properties of fiber bundles and fibrations. We will study their definitions, and constructions, while considering many examples. A main goal of these notes is to develop the topology needed to classify principal bundles, and to discuss various models of their classifying spaces. We will compute the cohomology of the classifying spaces of O(n) and U(n), and use them to study K - theory. These calculations will also allow us to describe characteristic v vi INTRODUCTION classes for these bundles which we use in a variety of applications including the study of framings, orientations, almost complex structures, Spin and SpinC - structures, vector fields, and immersions of manifolds. We will also discuss the Chern-Weil method of defining characteristic classes (via the curvature of a connection), and show how all U(n) - characteristic classes can be defined this way. We also use these techniques to consider the topological implications when a bundle admits a flat connection. Finally, we study the algebraic topology of fibrations. This includes the exact sequence in homotopy groups of a fibration, general obstruction theory, including the interpretation of characteristic classes as obstructions to the existence of cross sections, and the construction and properties of Eilenberg - MacLane spaces. We then study the spectral sequence of a filtration and the Leray - Serre spectral sequence for a fibration. A variety of applications are given, including the Hurewicz theorem in homotopy theory, and a calculation of the homology of certain loop spaces and Lie groups. None of the results in these notes are new. This is meant to be an exposition of classical topics that are of importance to students in any area of topology and geometry. There are several good references for many of the topics covered here, in particular the classical text of Steenrod on fiber bundles (describing the classification theorem) [39], the book of Milnor and Stasheff on characteristic classes of vector bundles [31], the texts of Whitehead [42] and Mosher and Tangora [32] for the results in homotopy theory. When good references are available we may not include the details of all the proofs. These notes grew out of a graduate topology course I gave at Stanford University during the Spring term, 1998. I am very grateful to the students in that course for comments on earlier versions of these notes. R.L. Cohen Stanford University August, 1998 CHAPTER 1 Locally Trival Fibrations In this chapter we define our basic object of study: locally trivial fibrations, or “fiber bundles”. We discuss many examples, including covering spaces, vector bundles, and principal bundles. We also describe various constructions on bundles, including pull-backs, sums, and products. We then study the homotopy invariance of bundles, and use it in several applications. Throughout all that follows, all spaces will be Hausdorff and paracompact. 1. Definitions and examples Let B be connected space with a basepoint b0 ∈ B, and p : E → B be a continuous map. Definition 1.1. The map p : E → B is a locally trivial fibration, or fiber bundle, with fiber F if it satisfies the following properties: −1 (1) p (b0) = F (2) p : E → B is surjective (3) For every point x ∈ B there is an open neighborhood Ux ⊂ B and a “fiber preserving −1 homeomorphism” ΨUx : p (Ux) → Ux ×F , that is a homeomorphism making the following diagram commute: −1 ΨUx p (Ux) −−−−→ Ux × F ∼= p proj y y Ux = Ux Some examples: • The projection map X × F −→ X is the trivial fibration over X with fiber F . 1 1 1 1 • Let S ⊂ C be the unit circle with basepoint 1 ∈ S . Consider the map fn : S → S given n 1 1 by fn(z) = z . Then fn : S → S is a locally trivial fibration with fiber a set of n distinct points (the nth roots of unity in S1). • Let exp : R → S1 be given by exp(t) = e2πit ∈ S1. Then exp is a locally trivial fibration with fiber the integers Z. 1 2 1. LOCALLY TRIVAL FIBRATIONS • Recall that the n - dimensional real projective space RPn is defined by n n RP = S / ∼ where x ∼ −x, for x ∈ Sn ⊂ Rn+1. Let p : Sn → RPn be the projection map. This is a locally trivial fibration with fiber the two point set. • Here is the complex analogue of the last example. Let S2n+1 be the unit sphere in Cn+1. Recall that the complex projective space CPn is defined by n 2n+1 CP = S / ∼ where x ∼ ux, where x ∈ S2n+1 ⊂ Cn, and u ∈ S1 ⊂ C. Then the projection p : S2n+1 → CPn is a locally trivial fibration with fiber S1. • Consider the Moebeus band M = [0, 1] × [0, 1]/ ∼ where (t, 0) ∼ (1 − t, 1). Let C be the “center circle” C = {(1/2, s) ∈ M} and consider the projection p : M → C (t, s) → (1/2, s). This map is a locally trivial fibration with fiber [0, 1]. Given a fiber bundle p : E → B with fiber F , the space B is called the base space and the space E is called the total space. We will denote this data by a triple (F, E, B). Definition 1.2. A map (or “morphism”) of fiber bundles Φ : (F1,E1,B1) → (F2,E2,B2) is a ¯ pair of basepoint preserving continuous maps φ : E1 → E2 and φ : B1 → B2 making the following diagram commute: φ¯ E1 −−−−→ E2 p1 p2 y y B1 −−−−→ B2 φ Notice that such a map of fibrations determines a continuous map of the fibers, φ0 : F1 → F2.