Existence theorem for coverings of Serre bundles

Coverings in the category of Serre bundles are investigated. A covering mapping of one such bundle onto another is understood as a morphism of the indicated category, consisting 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. It was also shown that the conjugacy class of this subsequence is an invariant of the corresponding covering. The main result of this paper is the existence theorem of a covering with a given invariant. In addition, it is proved here that the local triviality of the base bundle implies a similar property for the covering bundle.