Abstract
In this paper, we construct and study a category of principal fiber bundles with the following properties: 1) The base of a bundle is a polyhedron and the structure group is a k-dimensional torus. 2) The transition functions of the bundle atlas are smooth on simplexes of the base. 3) On the base, a simplicial action of a finite group Δ is given which has a multivalued lifting to the total space of the bundle. We study invariant connections and construct integer-valued realizable characteristic classes.
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