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Geometry and Topology Geometry and Topology Contents 14 Vector bundle basics 14{1 14.1 Introduction............................................ 14{1 14.2 Definitions and Examples.................................... 14{2 14.3 Examples............................................. 14{2 14.4 The tangent bundle........................................ 14{4 14.4.1 Submanifolds and the normal bundle.......................... 14{5 14.5 Operations on vector bundles.................................. 14{5 14.5.1 Morphisms, sub-bundles, quotient bundles, pull-back................. 14{5 14.5.2 Homotopy property of vector bundles......................... 14{6 14.6 Riemannian and Hermitian structures on vector bundles................... 14{6 15 Connections on vector bundles 15{1 15.1 Introduction............................................ 15{1 15.2 Definitions and examples..................................... 15{2 15.2.1 Description in terms of local data............................ 15{3 15.2.2 Existence of connections................................. 15{5 15.2.3 Pull-back......................................... 15{5 15.3 Connections and inner products................................. 15{5 16 Curvature 16{1 16.1 Parallel transport......................................... 16{1 16.2 Curvature of a connection.................................... 16{2 16.3 Curvature in terms of parallel transport............................ 16{5 17 Characteristic classes 17{1 17.1 Introduction............................................ 17{1 17.2 The first Chern class....................................... 17{2 17.2.1 First Chern class of a line bundle............................ 17{2 17.2.2 First Chern class of higher rank bundles........................ 17{2 17.3 Higher Chern classes....................................... 17{4 17.3.1 Traces of powers of the curvature............................ 17{4 17.3.2 Digression on invariant functions and symmetric functions.............. 17{5 17.3.3 The Chern class and the Chern character....................... 17{7 17.4 Pontryagin Classes........................................ 17{11 17.5 A serious calculation....................................... 17{12 17.6 The Euler class and the Pfaffian................................ 17{14 17.7 Gauss-Bonnet and Poincar´e-Hopf................................ 17{15 17.8 A theorem of Bott on foliations and characteristic classes.................. 17{21 Part C of the Geometry/Topology stream will be organized as follows. These notes cover the essential matter of this part of the course, and contain the important definitions, examples and theorems. The lectures themselves will not be an exercise in reading out these notes, but will back them up with a more detailed treatment of examples and discussion of the proofs. 0{1 (i) A large part of this material is covered in the excellent book of Bott and Tu. Unfortunately they do not treat connections and curvature, but for almost everything else it is still the best available source on this subject. These notes owe a great deal to that book. Geometry and Topology Lecture 14: Vector bundle basics after Michael Singera Contents 14.1 Introduction..................................... 14{1 14.2 Definitions and Examples............................. 14{2 14.3 Examples....................................... 14{2 14.4 The tangent bundle................................. 14{4 14.4.1 Submanifolds and the normal bundle........................ 14{5 14.5 Operations on vector bundles........................... 14{5 14.5.1 Morphisms, sub-bundles, quotient bundles, pull-back............... 14{5 14.5.2 Homotopy property of vector bundles....................... 14{6 14.6 Riemannian and Hermitian structures on vector bundles.......... 14{6 14.1 Introduction In essence a vector bundle is a continuously varying family of vector spaces, parameterized by some additional variables. This apparently abstract notion is of great importance in geometry and topology, and is the correct setting for gauge theory of mathematical physics. Here is a typical example. Suppose X is a topological space and we have a given continuous map Φ: X ! Hom(Rm; Rn). In other words, for each x 2 X, Φ(x) is an m×n matrix, depending continuously m upon x. Then for each x 2 X we have Kx ⊂ R , the kernel of Φx. As x varies in X, Kx should vary continuously, and indeed this is the case provided that the rank of Φx does not change. If we suppose that the rank of Φ(x) does not depend upon x 2 X, then the vector spaces Kx fit together to form a vector bundle over X. We shall explain precisely what is meant by ‘fit together to form a vector bundle' later in this lecture. There may not exist any obvious vector bundles over a general topological space. However, if M is a smooth manifold of dimension n, there is a natural vector bundle called the tangent bundle, denoted TM. For each point x, this can be pictured as the vector space of dimension n which `approximates M at x'. (This makes sense since x has a neighbourhood diffeomorphic to Rn.) TM was introduced in lecture 3 of this course, though not referred to there as a vector bundle. Given a vector bundle (which we continue to think of informally as a family of vector spaces Vx parame- terized by a space X) it is natural to ask whether one can choose, for each x 2 X, a basis (e1(x); : : : ; em(x)) of Vx whose elements depend continuously on x. The answer in general is no. If, however, the answer is `yes' then we say that the vector bundle is trivial. We shall see in subsequent lectures how to define topological invariants (characteristic classes) living in the (deRham) cohomology of X, with the property that if the vector bundle is trivial, then all characteristic classes must vanish. For simplicity in what follows, I shall work in the category of smooth manifolds (or smooth manifolds with boundary) unless otherwise stated. Most of the results remain true, but with perhaps different proofs, in the topological category as well. a [email protected] 14{1 14{2 14.2 Definitions and Examples How do we formalize the notion of a `smoothly varying family of vector spaces'? It is convenient to introduce a more general notion and then specialize. Definition 14.2.1 Let π : Z ! M be a smooth map from the manifold Z to the manifold M. We say that the data (Z; π) define a smooth fibre bundle, with typical fibre F , if there is a covering of M by open sets Uα with the following property: for each α, there is a diffeomorphism −1 φα : Uα × F ! π (Uα) (14.1) such that the first projection pr1 : Uα × F ! Uα agrees with the composite π ◦ φα. The space Z is referred to as the total space of the fibre bundle, M is called the base. This definition is supposed to capture the idea of a smoothly varying family of manifolds, each diffeo- morphic to F . For it follows from the definition that for each x 2 M, the subset π−1(x) of Z is a smooth submanifold of Z diffeomorphic to F , though not in a canonical fashion. We now specialize to define vector bundles. Definition 14.2.2 A smooth fibre bundle (E; π : E ! M) with typical fiber Rk is called a k-dimensional real vector bundle if (i) for each x 2 M, the set π−1(x) is equipped with the structure of a k-dimensional real vector space and (ii) the diffeomorphisms φα can be chosen in such a way that for each point x 2 Uα, −1 the bijective map z 7! φα(x; z) 2 π (x) is linear. Variations: complex k-dimensional vector bundle E ... typical fiber Ck ... Instead of k-dimensional vector bundle, we also say vector bundle of rank k. If the rank is equal to 1, then E is also called a (real or complex) line bundle. A (smooth) section of a fibre bundle π : Z ! M is a (smooth) map s : M ! Z with the property that −1 π ◦ s = idM . In other words, s(x) lies in the fibre of π (x) for every x 2 M. Strictly speaking we defined a vector bundle as a map π : E ! M, and the source of that map is the total space of the vector bundle. However it is extremely common to say and write the vector bundle E. Definition 14.2.3 Let π : E ! M and π0 : E0 ! M be smooth vector bundles. A smooth map f : E ! E0 is called a morphism of vector bundles if it commutes with the projections (π0 = π ◦ f) and 0 is linear on the fibres (fx : Ex ! Ex is linear for every x 2 M). The morphism is said to be injective (surjective) if each of the maps fx is injective (surjective). The morphism is called an isomorphism if it is both injective and surjective. Definition 14.2.4 The data E = M × V and π = pr1 define a vector bundle over M called the product bundle (with fibre V ). A vector bundle isomorphic to a product bundle is called a trivial vector bundle. One of the main goals of the (topological) study of vector bundles is to identify topological invariants which detect the non-triviality of vector bundles. Now, however, it is high time for some examples. 14.3 Examples This section is deliberately short on detail. The detail will be supplied in the first lecture. Example 14.3.1 (M¨obiusbundle) This is a real line-bundle over the circle S1 = R=Z. It can be defined as the quotient (orbit space) of R×R by the action of Z on R×R given by n·(x; y) = (x+n; −y). It can also be defined in terms of local data as follows. Let U0 = {−π < θ < πg, U1 = f0 < θ < 2πg. 1 Then U0 \ U1 = f0 < θ < πg [ fπ < θ < 2πg. To specify a line-bundle over S = U0 [ U1, it is enough ∗ to write down a smooth map from U0 [ U1 into GL(1; R) = R . We take this map to be identically equal to 1 on f0 < θ < πg and identically equal to −1 on fπ < θ < 2πg.
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