Surface braid groups, finite Heisenberg covers and double Kodaira fibrations

We exhibit new examples of double Kodaira fibrations using finite Galois covers of a product Σb x Σb, where 6b is a smooth projective curve of genus b " 2. Each cover is obtained by providing an explicit group epimorphism from the pure braid group P2(6b) 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 6b is at least ω(b + 1), where ω: N → 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.