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VECTOR BUNDLES AND THEIR APPLICATIONS VECTOR BUNDLES AND THEIR APPLICATIONS Glenys LUKE Oxford University Oxford Alexander MISHCHENKO Moscow State University Moscow KLUWER ACADEMIC PUBLISHERS Boston/London/Dordrecht CONTENTS v PREFACE In the last few years the use of geometric methods has permeated many more branchesofmathematicsandthesciences. Brieflyitsrolemaybecharacterized asfollows. Whereas methods of mathematical analysis describe phenomena ‘in the small’, geometric methods contribute to giving the picture ‘in the large’. A second no less important property of geometric methods is the convenience of using its language to describe and give qualitative explanations for diverse mathematical phenomena and patterns. From this point of view, the theory of vector bundles together with mathematical analysis on manifolds (global analysis and differential geometry) has provided a major stimulus. Its language turned out to be extremely fruitful: connections on principal vector bundles (in terms of which various field theories are described), transformation groups including the various symmetry groups that arise in connection with physical problems, in asymptotic methods of partial differential equations with small parameter, in elliptic operator theory, in mathematical methods of classical mechanics and in mathematical methods in economics. There are other cur- rently less significant applications in other fields. Over a similar period, uni- versityeducationhaschangedconsiderablywiththeappearanceofnewcourses ondifferentialgeometryandtopology. Newtextbookshavebeenpublishedbut ‘geometryandtopology’hasnot,inouropinion,beenwellcoveredfromaprac- tical applications point of view. Existing monographs on vector bundles have been mainly of a purely theoretical nature, devoted to the internal geometric andtopologicalproblemsofthesubject. Studentsfromrelateddisciplineshave found the texts difficult to use. It therefore seems expedient to have a simpler bookcontainingnumerousillustrationsandapplicationstovariousproblemsin mathematics and the sciences. Part of this book is based on material contained in lectures of the author, A.Mishchenko, given to students of the Mathematics Department at Moscow StateUniversityandisarevisedversionoftheRussianeditionof1984. Someof the less importanttheorems havebeen omitted and some proofs simplified and clarified. Thefocusofattentionwastowardsexplainingthemostimportantno- tionsandgeometricconstructionsconnectedwiththetheoryofvectorbundles. vii viii Vector bundles and their applications Theoremswerenotalwaysformulatedinmaximalgeneralitybutratherinsuch a way that the geometric nature of the objects came to the fore. Whenever possible examples were given to illustrate the role of vector bundles. Thus the bookcontainssectionsonlocallytrivialbundles,andonthesimplestproperties and operations on vector bundles. Further properties of a homotopic nature, including characteristic classes, are also expounded. Considerable attention is devoted to natural geometric constructions and various ways of constructing vector bundles. Basic algebraic notions involved in describing and calculating K-theory are studied and the particularly interesting field of applications to the theory of elliptic pseudodifferential operators is included. The exposition finisheswithfurtherapplicationsofvectorbundlestotopology. Certainaspects which are well covered in other sources have been omitted in order to prevent the book becoming too bulky. 1 INTRODUCTION TO THE LOCALLY TRIVIAL BUNDLES THEORY 1.1 LOCALLY TRIVIAL BUNDLES The definition of a locally trivial bundle was coined to capture an idea which recurs in a number of different geometric situations. We commence by giving a number of examples. The surface of the cylinder can be seen as a disjoint union of a family of line segmentscontinuouslyparametrizedbypointsofacircle. TheMöbiusbandcan be presented in similar way. The two dimensional torus embedded in the three dimensional space can presented as a union of a family of circles (meridians) parametrized by points of another circle (a parallel). Now, let M be a smooth manifold embedded in the Euclidean space RN and TM the space embedded in RN ×RN, the points of which are the tangent vectors of the manifold M. This new space TM can be also be presented as a unionofsubspacesT M,whereeachT M consistsofallthetangentvectorsto x x themanifoldM atthesinglepointx. ThepointxofM canbeconsideredasa parameter which parametrizes the family of subspaces T M. In all these cases x the space may be partitioned into fibers parametrized by points of the base. The examples considered above share two important properties: a) any two fibers are homeomorphic, b) despite the fact that the whole space cannot be presentedasaCartesianproductofafiberwiththebase(theparameterspace), if we restrict our consideration to some small region of the base the part of the fiber space over this region is such a Cartesian product. The two properties above are the basis of the following definition. 1 2 Chapter 1 Definition 1 Let E and B be two topological spaces with a continuous map p:E−→B. The map p is said to define a locally trivial bundle if there is a topological space F such that for any point x∈B there is a neighborhood U 3x for which the inverse image p−1(U) is homeomorphic to the Cartesian product U ×F. Moreover, it is required that the homeomorphism ϕ:U ×F−→p−1(U) preserves fibers, it is a ‘fiberwise’ map, that is, the following equality holds: p(ϕ(x,f))=x,x∈U,f ∈F. The space E is called total space of the bundle or the fiberspace , the space B is called the base of the bundle , the space F is called the fiber of the bundle and the mapping p is called the projection . The requirement that the homeomorphism ϕ be fiberwise means in algebraic terms that the diagram U ×F −ϕ→ p−1(U)     (cid:121)π (cid:121)p U = U where π :U ×F−→U, π(x,f)=x is the projection onto the first factor is commutative. One problem in the theory of fiber spaces is to classify the family of all locally trivial bundles with fixed base B and fiber F. Two locally trivial bundles p : E−→B and p0 : E0−→B are considered to be isomorphic if there is a homeomorphism ψ :E−→E0 such that the diagram E −ψ→ E0   (cid:121)p (cid:121)p0 B = B iscommutative. Itisclearthatthehomeomorphismψ givesahomeomorphism of fibers F−→F0. To specify a locally trivial bundle it is not necessary to be given the total space E explicitly. It is sufficient to have a base B, a fiber F and a family of mappings such that the total space E is determined ‘uniquely’ (up to isomorphisms of bundles). Then according to the definition of a locally trivial bundle, the base B can be covered by a family of open sets {U } such α