Sections of surface bundles

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 $d^2_{2,0}$ 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.Comment: 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 question

On Farber's invariants for simple 2q2q-knots

Let $K$ be a simple $2q$-knot with exterior $X$. We show directly how the Farber quintuple $(A,\Pi,\alpha,\ell,\psi)$ determines the homotopy type of $X$ if the torsion subgroup of $A=\pi_q(X)$ has odd order. We comment briefly on the possible role of the EHP sequence in recovering the boundary inclusion from the duality pairings $\ell$ and $\psi$. Finally we reformulate the Farber quintuple as an hermitian self-duality of an object in an additive category with involution.Comment: v2. Minor reorganization and corrections to final sectio

An indecomposable PD_3-complex : II

We show that there are two homotopy types of PD_3-complexes with fundamental group S_3*_{Z/2Z}S_3, and give explicit constructions for each, which differ only in the attachment of the top cell.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-47.abs.htm