Obstructions to stably fibering manifolds

TL;DR

This paper introduces algebraic K-theory-based obstructions to determine when a map between manifolds can be homotoped to a fiber bundle projection, providing criteria for stable fibering and extending results to Hilbert cube manifolds.

Contribution

It develops new algebraic K-theory obstructions for stable fibering of manifolds and generalizes fibering results to Hilbert cube manifolds.

Findings

Obstructions vanish when a map can be homotoped to a fiber bundle projection.

Provides criteria for the uniqueness of fibering.

Extends fibering results to Hilbert cube manifolds.

Abstract

Is a given map between compact topological manifolds homotopic to the projection map of a fiber bundle? In this paper obstructions to this question are introduced with values in higher algebraic K-theory. Their vanishing implies that the given map fibers stably. The methods also provide results for the corresponding uniqueness question; moreover they apply to the fibering of Hilbert cube manifolds, generalizing results by Chapman-Ferry.