# Surface braid groups, finite Heisenberg covers and double Kodaira fibrations

## TL;DR

This paper constructs new double Kodaira fibrations using finite Galois covers of product surfaces derived from pure braid groups and Heisenberg groups, revealing rich geometric structures and solutions to topology problems.

## Contribution

It introduces explicit constructions of double Kodaira fibrations via finite Heisenberg covers, expanding the class of known examples and linking algebraic groups to geometric topology.

## Key findings

- Every genus $b$ curve can serve as the base of a double Kodaira fibration.
- Number of non-isomorphic Kodaira fibred surfaces over a fixed curve grows with the number of prime factors.
- Constructed a real 4-manifold with signature 144 as a double surface bundle, solving a problem from Kirby's list.

## Abstract

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the arithmetic function counting the number of distinct prime factors of a positive integer. As a particular case of our general construction, we obtain a real $4$-manifold of signature $144$ that can be realized as a real surface bundle over a surface of genus $2$, with fibre genus $325$, in two different ways. This provides (to our knowledge) the first "double" solution to a problem from Kirby's list in low-dimensional topology.