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Allen Hatcher - Cornell University

Version , November 2017 Allen HatcherCopyrightc 2003 by Allen HatcherPaper or electronic copies for noncommercial use may be made freely without explicit permission from the other rights of ContentsIntroduction.. 1 Chapter 1. Vector Bundles.. Basic Definitions and Constructions.. 6 Sections 7. Direct Sums 9. Inner Products 11. Tensor Products Fiber Bundles Classifying Vector Bundles.. 18 Pullback Bundles 18. Clutching Functions 22. The UniversalBundle Structures on Grassmannians 31. Appendix: Paracompactness 2. K Theory.. The Functor K(X).. 39 Ring Structure 40. The Fundamental Product Theorem Bott Periodicity.. 51 Exact Sequences 51.

Introduction Everyone is familiar with the M¨obius band, the twisted product of a circle and a line, as contrasted with an annulus which is the actual product of a circle and a line.

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Transcription of Allen Hatcher - Cornell University

1 Version , November 2017 Allen HatcherCopyrightc 2003 by Allen HatcherPaper or electronic copies for noncommercial use may be made freely without explicit permission from the other rights of ContentsIntroduction.. 1 Chapter 1. Vector Bundles.. Basic Definitions and Constructions.. 6 Sections 7. Direct Sums 9. Inner Products 11. Tensor Products Fiber Bundles Classifying Vector Bundles.. 18 Pullback Bundles 18. Clutching Functions 22. The UniversalBundle Structures on Grassmannians 31. Appendix: Paracompactness 2. K Theory.. The Functor K(X).. 39 Ring Structure 40. The Fundamental Product Theorem Bott Periodicity.. 51 Exact Sequences 51.

2 Deducing Periodicity from the Product Theorem to a Cohomology Theory Division Algebras and Parallelizable Spheres.. 59H Spaces 59. Adams Operations 62. The Splitting Principle Bott Periodicity in the Real Case[not yet written] Vector Fields on Spheres[not yet written]Chapter 3. Characteristic Classes.. Stiefel-Whitney and Chern Classes.. 77 Axioms and Constructions 77. Cohomology of Grassmannians Euler and Pontryagin Classes.. 88 The Euler Class 91. Pontryagin Classes Characteristic Classes as Obstructions.. 98 Obstructions to Sections 98. Stiefel-Whitney Classes as Obstructions Classes as Obstructions 4. The J Homomorphism.. Lower Bounds on Im J.

3 108 The Chern Character 109. The e Invariant 111. Thom Spaces Denominators Upper Bounds on Im J[not yet written]Bibliography.. 119 PrefaceTopological K theory, the first generalized cohomology theory to be studied thor-oughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodic-ity Theorem of Bott proved just a few years earlier. In some respects K theory is moreelementary than classical homology and cohomology, and it is also more powerful forcertain purposes. Some of the best-known applications of algebraic topology in thetwentieth century, such as the theorem of Bott and Milnor that there are no divisionalgebras after the Cayley octonions, or Adams theorem determining the maximumnumber of linearly independent tangent vector fields on a sphere of arbitrary dimen-sion, have relatively elementary proofs using K theory, much simpler than the originalproofs using ordinary homology and first portion of this book takes these theorems as its goals, with an expositionthat should be accessible to bright undergraduates familiar with standard material inundergraduate courses in linear algebra, abstract algebra.

4 And topology. Later chap-ters of the book assume more, approximately the contents of astandard graduatecourse in algebraic topology. A concrete goal of the later chapters is to tell the fullstory on the stable J homomorphism, which gives the first level of depth in the stablehomotopy groups of spheres. Along the way various other topics related to vectorbundles that are of interest independent of K theory are also developed, such as thecharacteristic classes associated to the names Stiefel andWhitney, Chern, and is familiar with the M obius band, the twisted product of a circle and aline, as contrasted with an annulus which is the actual product of a circle and a bundles are the natural generalization of the M obius band and annulus, withthe circle replaced by an arbitrary topological space, called the base space of the vectorbundle, and the line replaced by a vector space of arbitrary finite dimension, called thefiber of the vector bundle.

5 Vector bundles thus combine topology with linear algebra,and the study of vector bundles could be called Linear Algebraic only two vector bundles with base space a circle and one-dimensional fiberare the M obius band and the annulus, but the classification of all the different vectorbundles over a given base space with fiber of a given dimensionis quite difficult ingeneral. For example, when the base space is a high-dimensional sphere and thedimension of the fiber is at least three, then the classification is of the same orderof difficulty as the fundamental but still largely unsolved problem of computing thehomotopy groups of the absence of a full classification of all the different vector bundles over agiven base space, there are two directions one can take to make some partial progresson the problem.

6 One can either look for invariants to distinguish at least some of thedifferent vector bundles, or one can look for a cruder classification, using a weakerequivalence relation than the natural notion of isomorphism for vector bundles. As ithappens, the latter approach is more elementary in terms of prerequisites, so let usdiscuss this is a natural direct sum operation for vector bundles over a fixed base spaceX, which in each fiber reduces just to direct sum of vector spaces. Using this, one canobtain a weaker notion of isomorphism of vector bundles by defining two vector bun-dles over the same base spaceXto be stably isomorphic if they become isomorphicafter direct sum with product vector bundlesX Rnfor somen, perhaps differentn s for the two given vector bundles.

7 Then it turns out that theset of stable isomor-phism classes of vector bundles overXforms an abelian group under the direct sumoperation, at least ifXis compact Hausdorff. The traditional notation for this groupisgKO(X). In the case of spheres the groupsgKO(Sn)have the quite unexpected prop-erty of being periodic inn. This is called Bott Periodicity, and the values ofgKO(Sn)are given by the following table:nmod 81 2 3 4 5 6 7 8gKO(Sn)Z2Z20Z0 0 0 ZFor example,gKO(S1)isZ2, a cyclic group of order two, and a generator for this groupis the M obius bundle. This has order two since the direct sum of two copies of the2 IntroductionM obius bundle is the productS1 R2, as one can see by embedding two M obius bandsin a solid torus so that they intersect orthogonally along the common core circle ofboth bands, which is also the core circle of the solid become simpler if one passes from real vector spaces to complex vectorspaces.

8 The complex version ofgKO(X), calledeK(X), is constructed in the same wayasgKO(X)but using vector bundles whose fibers are vector spaces overCrather thanR. The complex form of Bott Periodicity asserts simply thateK(Sn)isZfornevenand 0 fornodd, so the period is two rather than groupseK(X)andgKO(X)for varyingXshare certain formal properties withthe cohomology groups studied in classical algebraic topology. Using a more generalform of Bott periodicity, it is in fact possible to extend thegroupseK(X)andgKO(X)to a full cohomology theory, families of abelian groupseKn(X)andgKOn(X)forn Zthat are periodic innof period two and eight, respectively. There is more algebraicstructure here than just the additive group structure, however.

9 Tensor products ofvector spaces give rise to tensor products of vector bundles, which in turn give prod-uct operations in both real and complex K theory similar to cup product in ordinarycohomology. Furthermore, exterior powers of vector spacesgive natural operationswithin K all this extra structure, K theory becomes a powerful tool, in some waysmore powerful even than ordinary cohomology. The prime example of this is the verysimple proof, once the basic machinery of complex K theory has been set up, of thetheorem that there are no finite dimensional division algebras overRin dimensionsother than 1, 2, 4, and 8, the dimensions of the classical examples of the real andcomplex numbers, the quaternions, and the Cayley octonions.

10 The same proof showsalso that the only spheres whose tangent bundles are productbundles areS1,S3, andS7, the unit spheres in the complex numbers, quaternions, and classical problem that can be solved more easily using K theory thanordinary cohomology is to find the maximum number of linearlyindependent tangentvector fields on the sphereSn. In this case complex K theory is not enough, and theadded subtlety of real K theory is needed. There is an algebraic construction of therequisite number of vector fields using Clifford algebras, and the task is to show therecan be no more than this construction provides. Clifford algebras also give a niceexplanation for the mysterious sequence of groups appearing in the real form of let us return to the original classification problem for vector bundles over agiven base space and the question of finding invariants to distinguish different vectorbundles.


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