Approximately fibering a manifold over an aspherical one

TL;DR

This paper investigates when a closed connected manifold can be approximately fibered over an aspherical manifold, using algebraic K-theory obstructions and advanced topological theorems.

Contribution

It introduces algebraic K-theory obstructions for approximate fibrations and establishes conditions for their vanishing, linking several deep topological results.

Findings

Obstructions in algebraic K-theory determine approximate fibering.

Vanishing obstructions imply the existence of approximate fibrations.

Connections to Quinn's theorems and Farrell-Jones Conjectures are established.

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.