A new proof of the signature formula for surface bundles

Let E→X be an oriented surface bundle over a closed surface. Then the signature sign(E) is determined by the first Chern class of the flat vector bundle Γ associated to the monodromy homomorphism χ :π1(X)→Sp2h(Z) of E, it is equal to −4〈c1(Γ),[X]〉. The aim of this paper is to give an algebro-topological proof of this formula, i.e., one that does not use the Atiyah–Singer index theorem.