Abstract
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.
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