Sections of surface bundles

Abstract

A bundle with base $B$ and fibre $F$ aspherical closed surfaces has a section if and only if the action $:{\pi }_1(B)\to Out({\pi }_1(F))$ factors through $Aut({\pi }_1(F))$ and a cohomology class is 0. We simplify and make more explicit the latter condition. We also show that the transgression $d_{2,0}^2$ in the homology LHS spectral sequence of a central extension is evaluation of the extension class. Examples with hyperbolic fibre and no section (based on ideas of Endo) added.

Some new background material. Section on flat fibred case rewritten. In v3, details of Endo's example corrected and commentary added to final section on questions

References in corpus (1)

• $S^2$-bundles over 2-orbifolds

Cited by in corpus (2)

• Surface bundles and the section conjecture
• A note on the fundamental group of Kodaira fibrations