Introduction

• 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 base space) with some trivial bundle. 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,…