ABSTRACT. In [7,4], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n − 1)CP 2 #(2n − 1)CP 2 for each integer n ≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [7,4]. In particular, we construct (i) an infinitely many irreducible symplectic and non-symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n − 1)CP 2 #(2n − 1)CP 2 for each integer n ≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on BogomolovMiyaoka-Yau line with c 2 1 = 9χ h = 45, along with the exotic symplectic 4-manifolds constructed in [2,5,3,6,10].

We study the CP 2 -slicing number of knots, i.e. the smallest m ≥ 0 such that a knot K ⊆ S 3 bounds a properly embedded, null-homologous disk in a punctured connected sum (# m CP 2 ) × . We give a lower bound on the smooth CP 2 -slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth CP 2slicing number. We also give an upper bound on the topological CP 2 -slicing number in terms of the Seifert form and find knots for which the smooth and topological CP 2 -slicing numbers are both finite, nonzero, and distinct.

We introduce the 2-nodal spherical deformation of certain singular fibers of genus 2 fibrations, and use such deformations to construct various examples of simply connected minimal symplectic 4-manifolds with small topology. More specifically, we construct new exotic minimal symplectic 4-manifolds homeomorphic but not diffeomorphic to CP 2 #6CP 2 , CP 2 #7CP 2 , and 3CP 2 #kCP 2 for k = 16, 17, 18, 19 using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using g-nodal spherical deformations of certain singular fibers of genus g ≥ 3 fibrations.1991 Mathematics Subject Classification. Primary 57R55; Secondary 57R57, 57M05.

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We construct infinitely many distinct irreducible smooth structures on $n(S^2\,\times\,S^2)$, the connected sum of n copies of $S^2\,\times\,S^2$, for every odd integer $n\geq 27$.