Sections of Surface Bundles
Jonathan Hillman (Sydney)
Abstract
An \(F\)-bundle \(p:E\to{B}\) is a continuous map with fibres \(p^{-1}(b)\) homeomorphic to \(F\), for all \(b\in{B}\), and which is locally trivial: the base \(B\) has a covering by open sets \(U\) over each of which \(p\) is equivalent to the obvious projection of \(U\times{F}\) onto \(U\). (Thus \(p\) is a family of copies of \(F\), parametrized by \(B\).)
We shall assume that \(B\) and \(F\) are closed aspherical surfaces. Such bundles are then determined by the associated fundamental group extensions \[ 1\to\pi_1(F)\to\pi_1(E)\to\pi_1(B)\to1. \] We review this connection, and consider when such a bundle \(p\) has a section, i.e., a map \(s:B\to {E}\) such that \(ps=id_B\). If time permits we may say something about recent work by Nick Salter on the extent to which \(\pi_1(E)\) alone determines the bundle.