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Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover, we provide a method to compute the invariant for spaces over a K-formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander-Reiten theory for spaces due to Jorgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization. (C) 2011 Elsevier B.V. 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