We exhibit new examples of double Kodaira fibrations by using finite Galois
covers of a product Σb×Σb, where Σb 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(Σb)
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 Σb 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 list in low-dimensional topology.Comment: 38 pages, 3 figures. Final version, to appear in Annali della Scuola
Normale Superiore di Pisa, Classe di Scienz