Sections of surface bundles (Q906840)

The subject of the paper are surface bundles over surfaces, considering closed, aspherical surfaces as bases and fibers of the fibrations. To each such surface bundle is associated an extension of fundamental groups (and a cohomology extension class if the fiber is a torus, i.e. has abelian fundamental group), and the first part of the paper reviews in as far the algebra of such group extensions determines the possible bundles and fibrations (discussing e.g. fibrations of the same manifolds over different bases, geometric structures on the manifolds, fibrations over 2-orbifolds and virtual bundles, in the spirit of the author's monograph: [Four-manifolds, geometries and knots. Geometry and Topology Monographs 5. Coventry: Geometry \& Topology Publications (2002; Zbl 1087.57015)]). In a second part the author then concentrates on the existence of sections of such bundles. ``In particular, we simplify the cohomological obstruction, and show that the transgression \(d^2_{2,0}\) in the homology Lyndon-Hochschild-Serre spectral sequence of a central extension is evaluation of the extension class. We give also several examples of bundles without sections.'' For the entire collection see [Zbl 1333.57003].