# A note on the fundamental group of Kodaira fibrations

## TL;DR

This paper investigates the algebraic structure of the fundamental group of Kodaira fibrations, providing restrictions on possible groups and implications for symplectic 4-manifolds lacking Kähler structures.

## Contribution

It establishes new constraints on the fundamental groups of Kodaira fibrations based on coinvariant homology and relative irregularity, highlighting cases that cannot occur.

## Key findings

- If the fundamental group has relative irregularity g-s, then g ≤ 1 + 6s.
- Certain group extensions do not satisfy the derived bounds, thus cannot be fundamental groups of Kodaira fibrations.
- Examples of symplectic 4-manifolds without Kähler structures are identified through these restrictions.

## 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. \[ 1 \rightarrow \Pi_g \rightarrow \pi \rightarrow \Pi_b \rightarrow 1. \] 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 $m \colon \Pi_b \to \Gamma_g$ 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 $g \leq 1+ 6s$, 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 K\"ahler structure for reasons that eschew the usual obstructions.