The University of Sydney - School of Mathematics and Statistics

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 $$1\to {\pi }_1(F)\to {\pi }_1(E)\to {\pi }_1(B)\to 1.$$ We review this connection, and consider when such a bundle $p$ has a section, i.e., a map $s:B\to E$ such that $ps=i{d}_B$. 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.