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It is known that many simply connected, smooth topological $4$-manifolds admit
infinitely many smooth structures. The smaller the Euler characteristic, the
harder it is to construct exotic smooth structure.
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In this talk we present examples of symplectic 4-manifolds with same
integral cohomology as $S^2 \times S^2$. We also discuss the generalization of
these examples to $\#{2n-1}(S^2 \times S^2)$ for $n > 1$. As an application of these
symplectic building blocks, we construct exotic smooth structure on small 4-
manifolds such as $CP^2\# k(-CP^2)$ for $k = 2, 3, 4, 5$ and $3CP^2\# l(-CP^2)$ for
$l= 4, 5, 6, 7$. We will also discuss an interesting applications to the
geography of minimal symplectic $4$-manifolds.
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