Exotic rational elliptic surfaces without 1–handles

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.

Small exotic rational surfaces without 1- and 3-handles

“…(1) We can prove similarly to [15] that the Seiberg-Witten basic classes of X a,3 (resp. X a,3 ) are only ±K a,3 (resp.…”

“…(2) It later turned out that constructions of exotic rational surfaces in [15] In this section, we propose a strategy for rational blowdown constructions of exotic CP 2 #nCP 2 (1 ≤ n ≤ 9), though the author could not carry out the strategy for 1 ≤ n ≤ 4. Our constructions of exotic rational surfaces in [15] and Section 3 are based on the strategy.…”

“…Our constructions of exotic rational surfaces in [15] and Section 3 are based on the strategy. We begin with the question below.…”

“…Thus the following question seems to be reasonable: "What is the smallest n for which an exotic CP 2 #nCP 2 without 1-and 3-handles exists"? In [15], we constructed an exotic CP 2 #9CP 2 without 1-and 3-handles by using rational blowdowns and Kirby calculus. In [1], Akbulut later proved that the elliptic surface E(1) 2,3 , which is an exotic CP 2 #9CP 2 , has neither 1-nor 3-handles by using knot surgery and investigating a dual handle decomposition of E(1) 2,3 .…”

Elliptic surfaces without 1-handles

“…In this paper, we prove the following theorem by improving our previous procedure [7,8]. Our method is different from that of Akbulut.…”