A new proof of the signature formula for surface bundles (Q5937496)

The author gives a proof of the following theorem without using the Atiyah-Singer index theorem. Let \(E \to X\) be a smooth bundle whose fibre is an oriented closed surface of genus \(h \geq 2\) over an oriented closed surface \(X\) of genus \(g \geq 1\). Let \(\Gamma\) be the flat symplectic vector bundle associated to the monodromy homomorphism \(\chi : \pi (X) \to Sp_{2h}{\mathbf{Z}}\) of \(E\). Then \(\text{sign}(E)=-4\langle c_1(\Gamma),[X]\rangle\).