Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface requires both – and –handles. In this article, we construct a smooth –manifold which has the same Seiberg–Witten invariant as and admits neither – nor –handles by using rational blow-downs and Kirby calculus. Our manifold gives the first example of either a counterexample to the Harer–Kas–Kirby conjecture or a homeomorphic but nondiffeomorphic pair of simply connected closed smooth –manifolds with the same nonvanishing Seiberg–Witten invariants.
"Exotic rational elliptic surfaces without $1$–handles." Algebr. Geom. Topol. 8 (2) 971 - 996, 2008. https://doi.org/10.2140/agt.2008.8.971