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Abstract
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Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface
requires
both – and
–handles. In this article, we
construct a smooth –manifold
which has the same Seiberg–Witten invariant as
and admits
neither – nor
–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
–manifolds
with the same nonvanishing Seiberg–Witten invariants.
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Keywords
Kirby calculus, rational blow-down, 1-handle,
Seiberg–Witten invariant, small exotic 4-manifold
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Mathematical Subject Classification 2000
Primary: 57R55
Secondary: 57R65, 57R57, 57N13
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Publication
Received: 20 September 2007
Accepted: 5 December 2007
Published: 5 July 2008
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