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

Yasui, Kouichi¹

¹ Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan

Geometry & topology, Tome 23 (2019) no. 5, pp. 2685-2697.

Résumé

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 $b_2^{-}\not\equiv 1\left(mod4\right)$ 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.

Détail

DOI : 10.2140/gt.2019.23.2685

Classification : 57R55, 57R17, 57R65
Keywords: $4$–manifolds, handle decompositions, stable cohomotopy Seiberg–Witten invariants, symplectic structures

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