Obstructions to Fibering a Manifold

TL;DR

This paper introduces torsion obstructions that determine when a map between closed topological manifolds can be homotopy equivalent to a fiber bundle projection, extending Farrell's work for the case N = S^1.

Contribution

It defines new torsion obstructions for fibering manifolds and relates them to Farrell's obstructions when the fiber is a circle.

Findings

Torsion obstructions are necessary for a map to be homotopy equivalent to a fiber bundle projection.

When the fiber is S^1, the obstructions coincide with Farrell's obstructions.

The paper refines the understanding of fibering obstructions in topological manifolds.

Abstract

Given a map f: M \to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposition according to the comments of the referee and corrected some typos. The paper will appear in Geometriae Dedicata.