On genera of coverings of torus bundles

TL;DR

This paper investigates the structure of coverings of torus bundles over the circle, demonstrating that certain types of coverings do not reduce the Heegaard genus except in specific cases, thus clarifying the relationship between coverings and topological complexity.

Contribution

It proves that power coverings do not decrease the Heegaard genus of torus bundles and identifies conditions under which fiber coverings can lower the genus.

Findings

Power coverings do not lower Heegaard genus in torus bundles.

Fiber coverings lower the genus only in special cases.

Covering space factors into fiber and power coverings.

Abstract

After showing that a covering space of surface bundles over $S^1$ factors as a `covering of fibers' followed by a `power covering', we prove that, for torus bundles, power coverings do not lower Heegaard genus, and that fiber coverings lower the genus only in special cases.