A Construction of Surface Bundles over Surfaces with Non-zero Signature

Let Σg (respectively Σ/J be a closed oriented surface of genus g (respectively h), where g (respectively h) is a non-negative integer. Let Diff+Σh be the group of all orientation-preserving difϊeomorphisms of Σ/j with C°°-topology. A Σ^bundle over Σg (also called a surface bundle over a surface) is fiber bundle ξ = (£", Σg,p, Σ/^Diff+Σfr) over Σ^ with total space E, fiber Σ&, projection p : E —> Σg and structure group Diff+Σh. Our main concern is the signature τ(E) of the total space E of ξ. It is easily seen that if ξ is a trivial bundle then r(E) = τ(Σg)r(Σh) — 0. ChernHirzebruch-Serre [5] proved that if the fundamental group 7r(Σ5) of Σg acts trivially on the cohomology ring H*(Σh]R) of Σ^ then r(E) = 0. Kodaira [121 and Atiyah [1] gave examples of surface bundles over surfaces with non-zero signature. For each pair (m,t) of integers m,t G Z(m > 2,£ > 3), Kodaira constructed a surface bundle ξ = f (ra, t) with