  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 J{\o}rgensen. 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.
