Allen Hatcher - Vector Bundles and K-Theory

Allen Hatcher - Vector Bundles and K-Theory - Version 1.3...

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Version 1.3, July 2001 Allen Hatcher Copyright c ± 2001 by Allen Hatcher Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.

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Table of Contents Chapter 1. Vector Bundles 1.1. Basic Deﬁnitions and Constructions ............ 1 Sections 3. Direct Sums 5. Pullback Bundles 5. Inner Products 7. Subbundles 8. Tensor Products 9. Associated Bundles 11. 1.2. Classifying Vector Bundles ................. 1 2 The Universal Bundle 12. Vector Bundles over Spheres 16. Orientable Vector Bundles 21. A Cell Structure on Grassmann Manifolds 22. Appendix: Paracompactness 24. Chapter 2. Complex K-Theory 2.1. The Functor K(X) ....................... 2 8 Ring Structure 31. Cohomological Properties 32. 2.2. Bott Periodicity ........................ 3 9 Clutching Functions 38. Linear Clutching Functions 43. Conclusion of the Proof 45. 2.3. Adams’ Hopf Invariant One Theorem ........... 4 8 Adams Operations 51. The Splitting Principle 55. 2.4. Further Calculations ..................... 6 1 The Thom Isomorphism 61. Chapter 3. Characteristic Classes 3.1. Stiefel-Whitney and Chern Classes 6 4 Axioms and Construction 65. Cohomology of Grassmannians 70. Applications of w 1 and c 1 73. 3.2. The Chern Character 7 4 The J–Homomorphism 77. 3.3. Euler and Pontryagin Classes ................ 8 4 The Euler Class 88. Pontryagin Classes 91.
1. Basic Deﬁnitions and Constructions Vector bundles are special sorts of ﬁber bundles with additional algebraic struc- ture. Here is the basic deﬁnition. An n dimensional vector bundle is a map p : E B together with a real vector space structure on p - 1 (b) for each b B , such that the following local triviality condition is satisﬁed: There is a cover of B by open sets U α for each of which there exists a homeomorphism h α : p - 1 (U α ) U α × R n taking p - 1 (b) to { b R n by a vector space isomorphism for each b U α . Such an h α is called a local trivialization of the vector bundle. The space B is called the base space , E is the total space , and the vector spaces p - 1 (b) are the ﬁbers . Often one abbrevi- ates terminology by just calling the vector bundle E , letting the rest of the data be implicit. We could equally well take C in place of R as the scalar ﬁeld here, obtaining the notion of a complex vector bundle . If we modify the deﬁnition by dropping all references to vector spaces and replace R n by an arbitrary space F , then we have the deﬁnition of a ﬁber bundle: a map p : E B such that there is a cover of B by open sets U α for each of which there exists a homeomorphism h α : p - 1 (U α ) U α × F taking p - 1 (b) to { b F for each b U α . Here are some examples of vector bundles: (1) The product or trivial bundle E = B × R n with p the projection onto the ﬁrst factor. (2) If we let E be the quotient space of I × R under the identiﬁcations ( 0 ,t) ( 1 , - t) , then the projection I × R I induces a map p : E S 1 which is a 1 dimensional vector bundle, or line bundle . Since E is homeomorphic to a M¨obius band with its boundary circle deleted, we call this bundle the obius bundle .

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This note was uploaded on 03/09/2010 for the course MECHANICAL ME9802701 taught by Professor Prof.william during the Spring '10 term at Institut Teknologi Bandung.

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Allen Hatcher - Vector Bundles and K-Theory - Version 1.3...

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