Volume 23, issue 5 (2019)

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

Kouichi Yasui

Geometry & Topology 23 (2019) 2685–2697
Abstract

We show that every positive definite closed 4–manifold with b2+ > 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 b2+1 and b21(mod4) 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.

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