On fibering and splitting of 5-manifolds over the circle

TL;DR

This paper generalizes a 5-dimensional splitting theorem and investigates fibering and splitting of certain 5-manifolds over the circle, especially those with complex fiber structures and non-trivial fundamental groups.

Contribution

It extends Cappell's splitting theorem to broader classes of 5-manifolds and analyzes their fibering properties using topological cobordism techniques.

Findings

Generalized Cappell's 5-dimensional splitting theorem.

Analyzed smoothable splitting and fibering problems for specific 5-manifolds.

Identified limitations of existing surgery techniques for complex fibers.

Abstract

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.