Exotic rational elliptic surfaces without 1–handles

Yasui, Kouichi

doi:10.2140/agt.2008.8.971

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Exotic rational elliptic surfaces without $1$–handles

Kouichi Yasui

Algebraic & Geometric Topology 8 (2008) 971–996

DOI: 10.2140/agt.2008.8.971

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

Volume 8, issue 2 (2008)