#### 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 ${b}_{2}^{+}>1$ 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 ${b}_{2}^{+}\not\equiv 1$ and ${b}_{2}^{-}\not\equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$ 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