Declaration

I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Andrew John Korzeniewski) In 1957 Chern, Hirzebruch and Serre proved that the signature of the total space of a fibration of manifolds is equal to the product of the signatures of the base space and the fibre space if the action of the fundamental group of the base space on the real cohomology of the fibre is trivial. In the late 1960s Kodaira, Atiyah and Hirzebruch independently discovered examples of fibrations of manifolds with non-multiplicative signature. These examples are in the lowest possible dimension where the base and fibre spaces are both surfaces. W. Meyer investigated this phenomenon further and in 1973 proved that every multiple of four occurs as the signature of the total space of a fibration of manifolds with base and fibre both surfaces. Then in 1998 H. Endo showed that the simplest example of such a fibration with non-multiplicative signature occurs when the genus of the bas