Kodaira fibrations and mapping class groups

Kodaira fibrations are topological surface-by-surface bundles with infinite monodromy which admit a complex structure. The first examples were constructed by Kodaira and by Atiyah with the purpose of showing that the signature of manifolds is not multiplicative in fiber bundles. Since then the geometric and group theoretic properties of Kodaira fibrations have attracted much attention and formed the subject of many works. We will explain how they appear naturally when studying certain equivariant holomorphic maps between Teichmüller spaces. We will then use them to construct a virtual embedding of the mapping class group of a once punctured surface in the mapping class group of a closed surface of higher genus with irreducible image. This is joint work with Pierre Py.