  The paper is devoted to the problem when a map from some closed connected
manifold to an aspherical closed manifold approximately fibers, i.e., is
homotopic to Manifold Approximate Fibration. We define obstructions in
algebraic K-theory. Their vanishing is necessary and under certain conditions
sufficient. Basic ingredients are Quinn's thin h-Cobordism Theorem and End
Theorem, and knowledge about the Farrell-Jones Conjectures in algebraic K- and
L-theory and the MAF-Rigidity Conjecture by Hughes-Taylor-Williams.
