Monodromy of torus fiber bundles and decomposability problem

TL;DR

This paper introduces the concept of (stably) decomposable fiber bundles, focusing on torus fiber bundles over a circle, and provides a complete classification for low-dimensional cases.

Contribution

It offers a new framework for understanding decomposability of fiber bundles and characterizes stably decomposable torus bundles over a circle in low dimensions.

Findings

Characterization of stably decomposable torus bundles of dimension less than 4

Connection between decomposability and properties of matrices in SL(n, Z)

Complete classification in low-dimensional cases

Abstract

The notion of a (stably) decomposable fiber bundle is introduced. In low dimensions, for torus fiber bundles over a circle the notion translates into a property of elements of the special linear group of integral matrices. We give a complete characterization of the stably decomposable torus fiber bundle of fiber-dimension less than 4 over the circle.