Classification of orientable torus bundles over closed orientable surfaces

TL;DR

This paper classifies orientable torus bundles over closed orientable surfaces by analyzing group homomorphisms from surface fundamental groups to specific groups, leading to a comprehensive classification of such bundles and related extension problems.

Contribution

It provides a complete classification of these bundles using group homomorphisms up to mapping class group actions, extending previous results to more general groups.

Findings

Classified homomorphisms into PSL(2,Z) and SL(2,Z) up to mapping class group action.

Established that any orientable T^2-bundle over a surface with genus ≥ 1 is a fiber connected sum of T^2-bundles over T^2.

Extended classification to free products of finite cyclic groups and applied results to map extension problems.

Abstract

Let $g$ be a non-negative integer, $\Sigma _g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $\pi _1(\Sigma _g)\to G$ up to the action of $\mathcal{M}_g$ on $\pi _1(\Sigma _g)$ in the following cases; (1) $G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^2$-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable $T^2$-bundle over $\Sigma _g$ with $g\geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^2$-bundles over $T^2$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces.