## 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.