Almost principal bundles

The class $A$ of bundles with the following properties is investigated: each bundle in $A$ is the composition of a regular cover and a principal bundle (over the covering space) with Abelian structure group; the standard fibre $G$ of this decomposable bundle is a Lie group; the bundle has an atlas with multivalued transition functions taking values in the group $G$. The equivalence class of such an atlas will be called an almost principal bundle structure. The group of equivalence classes of almost principal bundles with a fixed base $B$ and a fixed structure group $G$ is computed, along with its subgroup of equivalence classes of principal $G$-bundles over $B$, and also the groups of equivalence classes of these bundles with respect to the morphisms of the category $C$ of decomposable bundles. A base and an invariant are found for an almost principal bundle that is not isomorphic to a principal bundle even in the category $C$. Applications are considered to the variational problem with fixed ends for multivalued functionals.