Commutative K-theory as a cohomology theory

Vector bundles over a compact manifold can be defined via transition 
functions to a linear group. Often one imposes 
conditions on this structure group. For example for real vector bundles on 
may  ask that all 
transition functions lie in the special orthogonal group to encode 
orientability. Commutative K-theory arises when we impose the condition 
that the transition functions commute with each other whenever they are 
simultaneously defined.

We will introduce commutative K-theory and some natural variants of it, 
and will show that they give rise to  new generalised 
cohomology theories.

This is joint work with Adem, Gomez and Lind building on previous work by 
Adem, F. Cohen, and Gomez.