Geometrically simply connected 4–manifolds and stable cohomotopy Seiberg–Witten invariants

Yasui, Kouichi

doi:10.2140/gt.2019.23.2685

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Geometrically simply connected $4$–manifolds and stable cohomotopy Seiberg–Witten invariants

Kouichi Yasui

Geometry & Topology 23 (2019) 2685–2697

DOI: 10.2140/gt.2019.23.2685

Abstract

We show that every positive definite closed $4$ –manifold with $b_{2}^{+} > 1$ and without $1$ –handles has a vanishing stable cohomotopy Seiberg–Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented $4$ –manifold with $b_{2}^{+} ≢ 1$ and $b_{2}^{-} ≢ 1 (mod 4)$ and without $1$ –handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the $1$ –handle condition, we prove these results under more general conditions which are much easier to verify.

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Keywords

$4$–manifolds, handle decompositions, stable cohomotopy Seiberg–Witten invariants, symplectic structures

Mathematical Subject Classification 2010

Primary: 57R55

Secondary: 57R17, 57R65

References

Publication

Received: 6 August 2018

Revised: 15 February 2019

Accepted: 21 April 2019

Published: 13 October 2019

Proposed: András I Stipsicz

Seconded: Ciprian Manolescu, Peter Ozsváth

Authors

Kouichi Yasui
Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University Suita Osaka Japan

Volume 23, issue 5 (2019)