Cocycles and Observers

Abstract

There exist two approaches to the theory of bundles, namely,

(i):

the observer approach based on cocycles, and

(ii):

the axiomatic geometric approach.

In (i), roughly speaking, we will use a cocycle in order to glue together product bundles. This will be considered in the present chapter. The geometric approach will be considered in the next chapter; this is the most elegant approach based on a few geometric axioms. For didactic reasons, we start with (i). The idea is to use bundle charts (product bundles) and to describe the change of bundle coordinates by a cocycle. In fact, the two approaches (i) and (ii) are equivalent to each other. In (ii), the cocycle corresponds to the transition maps between the bundle charts.