The first occurrence of topology in the study of fibred surfaces is the Zeuthen-Segre formula, which I shall recall;
the second one is a consequence of the Castelnuovo De Franchis theorem, which shows the topological nature
of maps to curves of genus at least two.
Kodaira fibrations illustrate the principle that the index is not multiplicative for differentiable fibre bundles, and the theorems of Fujita and Arakelov
relate this property to Variation of Hodge structure.
There are important open questions on fibred surfaces, one of them being the Shafarevich conjecture.
I shall explain, starting from the BCD surfaces which provide examples where the monodromy on the flat unitary bundle
in the Fujita decomposition is of infinte order, some criterion which ensures that the universal covering of a fibred surface S
I shall then discuss joint work with Corvaja and Zannier, and partial results concerning commutators in the symplectic group and in the mapping class group.
These questions arise from the question of understanding the singular fibres in the mysterious Cartwright-Steger.
I shall then produce a simple example, of a fibration of the product of two curves of genus 2 (or more) onto an elliptic curve,
with only one irreducible singular fibre. This shows that the product of 4 homologically independent Dehn twists is a commutator in Map_g,
for g at least 9.