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Preprint Version 2.0, dated January 2003, planned book for 2009, PDF file 700kB, PS zip 1.8 MB

Abstract:

The author's statement about this preprinted book contents is: "Table of Contents : Introduction.\\ Chapter 1. Vector Bundles. 1. Basic Definitions and Constructions. Sections. Direct Sums. Inner Products. Tensor Products. Associated Fiber Bundles. \\ 2. Classifying Vector Bundles. Pullback Bundles. Clutching Functions. The Universal Bundle. Cell Structures on Grassmannians. Appendix: Paracompactness \\ Chapter 2. K-Theory. 1. The Functor K(X). Ring Structure. The Fundamental Product Theorem. 2. Bott Periodicity. Exact Sequences. Deducing Periodicity from the Product Theorem. Extending to a Cohomology Theory. Elementary Applications. 3. Division Algebras and Parallelizable Spheres. H-Spaces. Adams Operations. The Splitting Principle. 4. Bott Periodicity in the Real Case. [Not yet written, ?] 5. Vector Fields on Spheres. [Not yet written, ?] \\ Chapter 3. Characteristic Classes. 1. Stiefel-Whitney and Chern Classes. Axioms and Constructions. Cohomology of Grassmannians. 2. Euler and Pontryagin Classes. The Euler Class. Pontryagin Classes. Chapter 4. The J-Homomorphism. 1. Lower Bounds on Im J. The Chern Character. The e Invariant. Thom Spaces. Bernoulli Denominators. 2. Upper Bounds on Im J. [in progress?]... \\\\ First on the slate of things to do is to extend the proof of Bott periodicity from the complex case to the real case, following the argument in Atiyah's book, using Clifford algebras."

Rights:

Open access: Copyright @ Allan Hatcher, 2003, 2008 http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf

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