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

TL;DR

This paper proves that certain simply connected 4-manifolds with specific topological properties have vanishing stable cohomotopy Seiberg-Witten invariants, implying they cannot support symplectic structures, under various conditions.

Contribution

It establishes new vanishing results for stable cohomotopy Seiberg-Witten invariants in 4-manifolds with no 1-handles, broadening understanding of symplectic structure obstructions.

Findings

Positive definite 4-manifolds with b2+>1 and no 1-handles have vanishing invariants.

Certain 4-manifolds with specific b2+ and b2- conditions lack symplectic structures.

Results hold under more general conditions beyond initial assumptions.

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 $b_2^-\not\equiv 1\pmod{4}$ 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.