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

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 $b_2^{-}\not\equiv 1(\mathrm{mod}4)$ 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.

9 pages, exposition improved, to appear in Geometry & Topology

References in corpus (4)

• Graph cobordisms and Heegaard Floer homology
• Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number
• Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
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Cited by in corpus (3)

• Lectures notes on Heegaard Floer homology
• Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces
• The simple type conjecture for mod 2 Seiberg-Witten invariants