Vector Bundles And K Theory Springerlink

Part of the book series: Classics in Mathematics ((CLASSICS,volume 212)) In Chapter 4 we defined the notion of a fibre bundle (a locally trivial fibration); in this chapter we consider an important class of fibre bundles—those for which every fibre has the structure of... We show how equivalence classes of such vector bundles over a CW-complex can be used to define groups K*(X) in such a way that K* becomes a cohomology theory. This is a preview of subscription content, log in via an institution to check access. Tax calculation will be finalised at checkout Unable to display preview.

Download preview PDF. The plan is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. Here is a provisional Table of Contents. At present only about half of the book is in good enough shape to be posted online, approximately 120 pages. This is available as a pdf file here. (I have reformatted this with narrower margins for a better reading experience on devices like an iPad, but for a paper copy with more standard size margins try printing at 85-90 per cent of...

What is included in this installment is: Much of this material is already well covered in other sources, notably the classic books of Atiyah (K-theory) and Milnor & Stasheff (Characteristic Classes). These books are still in print, although they have become somewhat expensive. Eventually I intend for the book to include a few more things that aren't readily accessible elsewhere, such as the full story on the stable J homomorphism. What is posted now is Version 2.2, dated November 2017. This is a minor revision of Version 2.0 from January 2003, with the addition (in 2017) of a section 3.3 giving the obstruction theory definition of Stiefel-Whitney classes.

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Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Connect and share knowledge within a single location that is structured and easy to search. The definition of vector bundles seems to be split in the mathematical community: some sources insist that the rank of each fibre is the same, whereas some don't ask for this requirement. The classification of vector bundles with arbitary base space and different dimensions of fibers(vector spaces over real/complex numbers) in terms of isomorphism of vector bundles is a hard problem.

We can use topological invariants to distinguish non-isomorphic vector bundles, but this is far from complete. Another way is to weaken the classification, this is the idea of K theory, we try to classify the vector bundles in equivalence classes defined via Whitney sum(direct sum of fiber bundles over same... After this sum, the resulting fiber bundle are isomorphic, the summed trivial bundle may be different for different vector bundles. This is known as stable isomorphism The stable isomorphism equivalence classes form an abelian group under direct sum operation. There turns out to some kind of Bott periodicity.

With a more general form of Bott periodicity, it’s possible to extend the groups to a full cohomology theory, with more algebraic structures than just group structure. This makes K theory a powerful tool. Important classes of topological invariants are cohomological invariants, known as characteristic classes. Examples include Z2 coefficents: Stiefel-Whitney class(measuring orientability, refined sort of orientability:spin structure,…), Z coefficients: Pontryagin, Euler classes, Chern classes,…

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