Surface bundles with genus $2$ fibre are not negatively curved

Caterina Campagnolo

Published 2016-03-24Version 1

In the present note we prove that the total space of a $\Sigma_2$-bundle over a hyperbolic surface does not admit a negatively curved structure. This is a first step towards an answer to the question whether the total space of a surface bundle over a surface admits a hyperbolic structure. The proof relies on the result of Gromov that a word hyperbolic group does not contain a subgroup isomorphic to $\mathbb{Z}^2$, and on a geometric description of the second homology group of the total space of the bundle.