A note on the fundamental group of Kodaira fibrations

Abstract

The fundamental group $\pi$ of a Kodaira fibration is, by definition, the extension of a surface group ${\Pi }_b$ by another surface group ${\Pi }_g$, i.e. \[ 1 \rightarrow Π_g \rightarrow π\rightarrow Π_b \rightarrow 1. \] Conversely, we can inquire about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restriction on the image of the classifying map $m:{\Pi }_b\to {\Gamma }_g$ in terms of the coinvariant homology of ${\Pi }_g$. In particular, we observe that if $\pi$ is the fundamental group of a Kodaira fibration with relative irregularity $g-s$, then $g\le 1+6s$, and we show that this effectively constrains the possible choices for $\pi$, namely that there are group extensions as above that fail to satisfy this bound, hence cannot be the fundamental group of a Kodaira fibration. In particular this provides examples of symplectic $4$--manifolds that fail to admit a Kähler structure for reasons that eschew the usual obstructions.

7 pages, minor revision. To appear in Proc. Edinb. Math. Soc. (2)

References in corpus (1)

• Sections of surface bundles