TABLE OF CONTENTS Due to the de Rham Theorem 1.8.4, the characteristic classes of principal bundles are represented by the De Rham cohomology classes of certain exterior forms expressed in the strength of principal connections [142; 220; 333]. One meets these characteristic forms in many quantum field models.
Let G be a topological group. A locally trivial topological fibre bundle Y ? X with a typical fibre V is called a fibre bundle with a structure group G (or, simply, a G-bundle) if G acts effectively on V on the left and admits an atlas ? = {( U ?, ? ?), ???} whose transition functions ? ??( x) take their values in the group G. In view of the cocycle condition (10.6.2), the transition functions ? ?? of a G-bundle constitute a one-cocycle (1.7.12) (written in the multiplicative form) of the presheaf G({ U}) of G-valued continuous functions on X. Let ?' be another atlas over the same cover. Atlases ? and ?' are equivalent and characterize the same G-bundle if there is a 0-cochain { g ?} of the presheaf G({ U}) such that
This implies
| (10.10.1) |  |
Accordingly, the cocycles obeying the relation (10.10.1) axe said to be equivalent. The equivalence classes of cocycles associated to a fixed cover 
of X constitute the cohomology set
