Approximately fibering a manifold over an aspherical one

Abstract

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.

final version, published online in Math. Annalen, 46 pages, Added acknowledgement for the ERC Advanced Grant "KL2MG-interactions" (no. 662400) of the second author granted by the European Research Council