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Bott Periodicity Bott Periodicity David Manuel Murrugarra Tomairo Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Peter Haskell, Chair William Floyd Joseph Ball May 8, 2007 Blacksburg, Virginia Keywords: K-theory, Bott periodicity Copyright 2007, David Manuel Murrugarra Tomairo Bott Periodicity David Manuel Murrugarra Tomairo (ABSTRACT) Bott periodicity plays a fundamental role in the definition and un- derstanding of K-theory, the generalized cohomology theory defined by vector bundles. This paper examines the proof, given by Atiyah and Bott[3], of the periodicity theorem for the complex case. We also describe the long exact sequence for K-cohomology in the cat- egory of connected finite CW-complexes. Dedication To my mother: Eugenia Victoria Tomairo Geronimo. iii Acknowledgments I am very thankful to Dr. Peter Haskell for his support and patience while he was advising me this paper. Also, I would like to thank Dr. Ball and Dr. Floyd for reading this paper and making corrections. I had the pleasure of meeting wonderful people in the Math depart- ment. I have learned a lot of mathematics from my professors. I have also learned many things in math from my classmates and friends. I wish to thank Quanlei, Bart, Arturo, Moises, Carlos, etc. I am also very thankful to Ratchada, Alejandra and Jessica for their friendship. Finally, I am very grateful to my family for their support and love. iv Contents Introduction 1 1 Generalities on Vector Bundles 3 1.1 Basic Definitions and results . 3 1.2 Clutching Functions . 6 1.3 Clutching Construction for vector bundles over S2 . 8 1.4 Universal Bundles . 11 2 The K and Ke Cofunctors 13 2.1 The K Cofunctor . 13 2.2 The Cofunctor Ke ......................... 15 2.3 Representation of Ke(X) ..................... 16 2.4 The External Product . 17 3 Bott Periodicity 19 3.1 Some calculations . 19 3.2 Linearization . 23 3.3 Analysis of Linear Clutching Functions . 27 3.4 The inverse of the Periodicity Isomorphism for S2 . 32 3.5 The Periodicity Theorem . 34 4 K-Cohomology 35 4.1 Suspensions . 35 4.2 Homotopy Sequences . 36 4.3 Bott periodicity for the reduced case . 40 4.4 Extending to a generalized cohomology theory . 41 Bibliography 43 v Introduction The Bott periodicity theorem was discovered by Raoul Bott in 1950 by using elements of Morse theory[4]. After that, many others came up with dif- ferent proofs of Bott periodicity (see Husemoller[6], p.150). Bott periodicity plays a fundamental role in the definition and understanding of K-theory, the generalized cohomology theory defined by vector bundles. In 1964, Atiyah and Bott gave, as they say in[3], an “elementary proof” of the periodicity theorem. This thesis explains the techniques used by Atiyah and Bott in their proof. The Bott periodicity theorem can be formulated in many ways. One of the simplest ways to state the Bott Periodicity Theorem is the following: there is an explicit isomorphism between K(X) ⊗ K(S2) and K(X × S2) for a compact Hausdorff space X. We examine this version of the Bott pe- riodicity theorem, and also give a version the periodicity theorem for the reduced K-theory: Ke(X) is isomorphic to Ke(S2X). As a consequence of the last statement, in the last chapter, we show the degree-two periodicity of K-cohomology. The arrangement of the paper is as follows. We start by giving the preliminaries needed to construct the K-cofunctor. Thus in the first chapter we give the definition of a vector bundle and describe operations on vector bundles, such as direct sums and tensor products. We also describe a way to construct vector bundles via clutching functions. We finish the first chapter by speaking informally about the universal vector bundle associated with the unitary group. In the second chapter, we define the rings K(X) and Ke(X) for a compact Hausdorff space X. Here, we care only about the case of complex K-Theory. We describe the representation for Ke(X) in terms of a set of homotopy classes of maps from X to a special space BU, which is 1 defined in chapter two. In the third chapter, we calculate K(S2) by using the techniques used by Atiyah and Bott in the proof of the periodicity theorem. Even though we can also compute K(S2) directly by using the knowledge of the homotopy groups of the unitary groups, U(n)’s, we focus on the Atiyah- Bott approach because it brings into a better understanding of the techniques that are essential to the proof of the complete periodicity theorem. Indeed, the general proof of the periodicity theorem is a parametrized version of the proof for the 2-sphere case. Finally, in the fourth chapter, we start by defining the suspension operation over pointed spaces and then constructing homotopy exact sequences with this kind of operation over compact spaces. Then, by using these exact homotopy sequences and the representation of the cofunctor Ke, we are able to state the Bott periodicity theorem for the reduced Ke-theory case. We finish this chapter with a discussion of the long exact sequence for K-theory. 2 Chapter 1 Generalities on Vector Bundles In this first chapter we give the preliminaries needed to construct the K-cofunctor in the next chapter and eventually part of the machinery to be used to prove the periodicity theorem in the third chapter. In the first section we give the definition of a vector bundle and a description of how to obtain vector bundles via basic operations like direct sum and tensor products. In the second section we describe a way to construct vector bundles with base space a sphere via clutching functions. 1.1 Basic Definitions and results We start by giving the definition of a vector bundle and describing the basic operations between vector bundles, namely, direct sums and tensor products. The last part of this section has to do with the construction of a vector bundle induced by a function, i.e., the pull-back of a vector bundle under a given function. Definition 1.1.1. A k-dimensional vector bundle is a triple ξ = (E, p, X), where E and X are topological spaces and p : E → X is a function such that the following conditions are satisfied (a) For each x ∈ X, p−1(x) is a k-dimensional vector space. (b) Local triviality condition: Each x ∈ X has an open neighborhood and a homeomorphism h : U × Ck → p−1(U) such that the restriction x × Ck → p−1(x) is a vector space isomorphism. 3 4 The spaces E and X in the definition above are referred as the total space and the base space respectively. The vector space p−1(x) is called the fibre of the vector bundle at x ∈ X. Sometimes the space E of a vector bundle ξ, is denoted by E(ξ). In the next example we describe a vector bundle that we will use many times for our constructions. 1 Example 1.1.2. The canonical complex line bundle H is given by (E, p1, CP ), 1 2 where E = {(`, ν) ∈ CP × C : ν ∈ `} and p1(`, ν) = `. In the above notation CP 1 is the quotient of C2\{0} under the equivalence rela- 1 tion (z0, z1) ∼ λ(z0, z1) for λ ∈ C\{0}, so ` ∈ CP represents the line {λ(z0, z1): λ ∈ C\{0}}. Given two vector bundles ξ1 and ξ2, we can construct a new vector bundle ξ1 ⊕ ξ2 which has as a fiber the direct sum of the fibers of ξ1 and ξ2. We make this clearer in the following definition. Definition 1.1.3. The Whitney sum ξ1 ⊕ ξ2 of two vector bundles ξ1 = (E1, p1,X) and ξ2 = (E2, p2,X) is the triple (E1 ⊕ E2, q, X) where 0 0 E1 ⊕ E2 = {(x, x ) ∈ E1 × E2 : p1(x) = p2(x )} 0 and q(x, x ) = p1(x) = p2(x). From the above definition it is clear that q−1(x) = p−1(x) ⊕ p−1(x), therefore we have the following basic result. Lemma 1.1.4. The fibre of ξ1 ⊕ ξ2 over x ∈ X is the direct sum of the fibres of ξ1 and ξ2 over x. Similarly we can construct a vector bundle ξ1 ⊗ξ2 whose fiber is the tensor product of the fibers of ξ1 and ξ2. Definition 1.1.5. The tensor product ξ1 ⊗ ξ2 of two vector bundles ξ1 = (E1, p1,X) and ξ2 = (E2, p2,X) is the triple (E1 ⊗ E2, q, X) where E1 ⊗ E2 −1 −1 is the disjoint union of p1 (x) ⊗ p2 (x) for all x ∈ X and q is the canonical projection on each fibre. It is clear from the definition that the fibre of ξ1 ⊗ ξ2 over x ∈ X is −1 −1 p1 (x) ⊗ p2 (x). Given a vector bundle ξ over Y and a function f : X → Y , we can construct another vector bundle f ∗(ξ) over X. This vector bundle is called the pullback of ξ under f and we describe this construction in the following definition. 5 Definition 1.1.6. Let ξ = (E, p, Y ) be a vector bundle and f : X → Y ∗ 0 be a function. Then the pullback f (ξ) of ξ under f is the triple (E , p1,X) 0 where E = {(x, e) ∈ X × E/f(x) = p(e)} and p1 is the projection on the first factor. Definition 1.1.7. A morphism between two vector bundles ξ = (E, p, X) and η = (F, q, X) is a map f : E → F , that commutes with the projections p and q, i.e., qf = p, and that has the restriction f : p−1(x) → q−1(x) a linear map for all x ∈ X.
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