Existence Theorem for Coverings of Serre Bundles

In this paper, we study coverings of the Serre bundle category. A covering mapping of one such bundle onto another is understood as a morphism of the indicated category that consists of covering mappings of total spaces and bases. Earlier, the author associated each such covering with a subsequence of the homotopy sequence of the base bundle. The conjugacy class of this subsequence was also shown to be an invariant of the corresponding covering. The main result of this study is the existence theorem for a covering with a specified invariant. The local triviality of the base bundle is proved here to imply a similar property for the covering bundle.