On the levels of maps and topological realization of objects in a triangulated category

The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. 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 Jørgensen. 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.