Smooth almost $\Delta$-fiber bundles over simplicial complexes

In this paper we construct and study a category of principal fiber bundles with the following properties: 1) the base is a simplicial complex and the structure group is a $k$-dimensional torus, 2) maps of any atlas are smooth on every simplex of the base, and 3) the finite group $\Delta$ acts on the base and this action has a multi-valued lifting to the total space. We study invariant connections and built integer-valued realizable characteristic classes.