Smooth structures on 4-manifolds with small Euler characteristics
We investigate the exotic smooth structures on 4-manifolds with small Euler characteristic. Using the surgical techniques of knot surgery and fiber sum, we construct new symplectic 4-manifolds with zero signature and cohomology equivalent to S2 x S2. In fact, a more general construction provides exotic manifolds with zero signature and cohomology equivalent to #(29_1)(S2 x S2) for any g. Applying the surgical methods of Fintushel-Stern [FS4], we also construct simply connected nonsymplectic 4-manifolds starting from the elliptic surfaces E(n) for n > 3. Seiberg-Witten invariants are used to distinguish their smooth structures. «'e also give a uniform technique that yields non-symplectic manifolds homeomorphic CP 2#k CP 2 for k = 5, 6 and also 3CP 2#8 CP 2. The main construction used in our proofs are knot surgery and rational-blowdown introduced by Fintushel-Stern. Next we discuss surface bundles over surfaces with lion-zero signature. Using the Lefschetz fibration structure on the knot surgered elliptic surfaces E(n)h, recent result by Fintushel-Stern, and a gluing technique, we give a new construction that yields the infinitely many such surface bundles with non-zero signature.