Topological K-theory is a generalised cohomology theory on the category of compact Hausdorff spaces. It classifies the vector bundles over a space up to stable equivalences. Equivalently, via the Serre-Swan theorem, it classifies the finitely generated projective modules over the -algebra .
Let be a unital -algebra over and denote by the algebraic direct limit of matrix algebras under the embeddings . Identify the completion of with the stable algebra (where is the compact operators on ), which we will continue to denote by . The group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the projections in . Two projections and are homotopic if there exists a norm continuous path of projections from to . Let and be two projections. The sum of their homotopy classes and is the homotopy class of their direct sum: where . Alternatively, one can consider equivalence classes of projections up to unitary transformations. Unitary equivalence coincides with homotopy equivalence in (or for large enough).
Denote by the direct limit of unitary groups under the embeddings . Give the direct limit topology, i.e. a subset of is open if and only if is an open subset of , for all . The group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in . Two unitaries and are homotopic if there exists a norm continuous path of unitaries from to . Let and be two unitaries. The sum of their homotopy classes and is the homotopy class of their direct sum: where . Equivalently, one can work with invertibles in (an invertible is connected to the unitary via the homotopy ).
Higher K-groups can be defined through repeated suspensions,
But, the Bott periodicity theorem means that
The main properties of are:
There are three flavours of topological K-theory to handle the cases of being complex (over ), real (over ) or Real (with a given real structure).
Real K-theory has a Bott period of 8, rather than 2.
- 1 N. E. Wegge-Olsen, K-theory and -algebras. Oxford science publications. Oxford University Press, 1993.
- 2 B. Blackadar, K-Theory for Operator Algebras. Cambridge University Press, 2nd ed., 1998.
- 3 M. Rørdam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for -Algebras. Cambridge University Press, 2000.
|Date of creation||2013-03-22 12:58:06|
|Last modified on||2013-03-22 12:58:06|
|Last modified by||mhale (572)|