, Volume 181, Issue 3, pp 577-603

Exotic smooth structures on small 4-manifolds with odd signatures

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Abstract

Let M be \((2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2}\) for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \(\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2}\) or \(3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2}\) for k=6,8,10.