Exotic rational elliptic surfaces without $1$–handles

Abstract

Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E{\left(1\right)}_{2,3}$ requires both $1$– and $3$–handles. In this article, we construct a smooth $4$–manifold which has the same Seiberg–Witten invariant as $E{\left(1\right)}_{2,3}$ and admits neither $1$– nor $3$–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 $4$–manifolds with the same nonvanishing Seiberg–Witten invariants.