Existence of symplectic structures on torus bundles over surfaces

TL;DR

This paper establishes a criterion for when a torus bundle over a surface admits a symplectic structure, linking the existence of such structures to the nontriviality of the fiber's homology class, using advanced topological tools.

Contribution

It provides a complete characterization of symplectic structures on torus bundles over surfaces via Seiberg--Witten theory and spectral sequences.

Findings

A torus bundle over a surface admits a symplectic structure if and only if the fiber's homology class is nonzero.

The paper applies Seiberg--Witten invariants to classify symplectic structures on these bundles.

Spectral sequences are used to analyze the homological conditions for symplectic structures.

Abstract

Let E be the total space of a locally trivial torus bundle over the surface \Sigma_g of genus g>1. Using the Seiberg--Witten theory and spectral sequences we prove that E carries a symplectic structure if and only if the homology class of the fiber [T^2] is nonzero in H_2(E,R).