Volume 8, issue 2 (2008)

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Exotic rational elliptic surfaces without $1$–handles

Kouichi Yasui

Algebraic & Geometric Topology 8 (2008) 971–996
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 nondiffeomorphic pair of simply connected closed smooth 4–manifolds with the same nonvanishing Seiberg–Witten invariants.

Keywords
Kirby calculus, rational blow-down, 1-handle, Seiberg–Witten invariant, small exotic 4-manifold
Mathematical Subject Classification 2000
Primary: 57R55
Secondary: 57R65, 57R57, 57N13
References
Publication
Received: 20 September 2007
Accepted: 5 December 2007
Published: 5 July 2008
Authors
Kouichi Yasui
Department of Mathematics
Graduate School of Science
Osaka University
Toyonaka, Osaka 560-0043
Japan