Abstract. For every integer k ≥ 2, we construct infinite families of mutually nondiffeomorphic irreducible smooth structures on the topological 4-manifolds (2k − 1)(S 2 × S 2 ) and (2k − 1)(CP 2 #CP 2 ), the connected sums of 2k − 1 copies of S 2 × S 2 and CP 2 #CP 2 .

For each pair (e, σ) of integers satisfying 2e + 3σ ≥ 0, σ ≤ −2, and e + σ ≡ 0 (mod 4), with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic e and signature σ. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature −1) with Euler characteristic 4k (resp. 4k + 1) for all k ≥ 46 (resp. k ≥ 49).

Abstract. The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to CP 2 #(2k + 1)CP 2 for k = 1, . . . , 4, or to 3CP 2 #(2l + 3)CP 2 for l = 1, . . . , 6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on CP 2 #3CP 2 , 3CP 2 #5CP 2 and 3CP 2 #7CP 2 .

Abstract. For any pair of integers n ≥ 1 and q ≥ 2, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class q[F ] of an elliptic surface E(n), where [F ] is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic 4-manifolds.

Abstract. We present an algorithm that produces new families of closed simply connected spin symplectic 4-manifolds with nonnegative signature that are interesting with respect to the symplectic geography problem. In particular, for each odd integer q satisfying q ≥ 275, we construct infinitely many pairwise nondiffeomorphic irreducible smooth structures on the topological 4-manifold q(S 2 × S 2 ), the connected sum of q copies of S 2 × S 2 .