Exotic rational elliptic surfaces without 1-handles

TL;DR

This paper constructs a smooth 4-manifold with the same Seiberg-Witten invariant as a known elliptic surface, but without 1- or 3-handles, challenging existing conjectures about handle decompositions.

Contribution

It provides the first example of a 4-manifold that contradicts the Harer-Kas-Kirby conjecture or shows a pair of homeomorphic but non-diffeomorphic manifolds with identical invariants.

Findings

Constructed a 4-manifold without 1- or 3-handles

Manifold shares Seiberg-Witten invariants with $E(1)_{2,3}$

Challenges the Harer-Kas-Kirby conjecture

Abstract

Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ 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 non-diffeomorphic pair of simply connected closed smooth 4-manifolds with the same non-vanishing Seiberg-Witten invariants.