SURFACE BRAID GROUPS, FINITE HEISENBERG COVERS AND DOUBLE KODAIRA FIBRATIONS

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $Sigma_b imes Sigma_b$, 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 $oldsymbol{\upomega}(b+1)$, where $oldsymbol{\upomega} colon mathbb{N} o mathbb{N}$ 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 problem list in low-dimensional topology.