Geometrically simply connected 4-manifolds and stable cohomotopy Seiberg-Witten invariants (Q2331025)

A compact, connected smooth four-dimensional manifold is called geometrically simply connected, if it admits a handlebody decomposition without \(1\)-handles. As the \(1\)-handles provide generators, the vanishing of the fundamental group is a necessary condition for a manifold to be geometrically simply connected. It is an open question, whether it also is a sufficient condition in the case of a closed \(4\)-manifold. Indeed, this question belongs to a circle of inter-related problems around the -- as of now unsolved -- smooth Poincaré conjecture in \(4\) dimensions, cf. e.g. Kirby's Problem List [AMS/IP Stud. Adv. Math. 2, 35--473 (1997; Zbl 0888.57014)]. The main results of the article can be summarized in the following statement: Theorem. Let \(X\) be a geometrically simply connected, closed, oriented, smooth \(4\)-manifold. -- If the intersection form is positive definite with \(b_{2}^{+}>1\), then the stable cohomotopy Seiberg-Witten invariant of \(X\) vanishes. -- If \(b_{2}^{+}\not\equiv1\) and \(b_{2}^{-}\not\equiv1\) (mod \(4\)), then at least one of \(X\) and \(\overline{X}\) admits no symplectic structure. Here \(\overline{X}\) denotes the \(4\)-manifold \(X\) equipped with the reverse orientation. Note that \(S^{2}\times S^{2}\) admits a symplectic structure for either orientation. For the proof, the author observes that a \(2\)-handle neighborhood in \(X\) can be used to construct a homologically non-trivial, embedded \(2\)-sphere with self-intersection number zero in the connected sum \(X\#\overline{X}\). The result now follows from the connected sum theorem [\textit{S. Bauer}, Invent. Math. 155, No. 1, 21--40 (2004; Zbl 1051.57038)] for the stable cohomotopy Seiberg-Witten invariants, combined with a vanishing theorem [Geom. Topol. 9, 1--93 (2005; Zbl 1087.57022)] of \textit{K. A. Frøyshov} and a non-vanishing theorem [Math. Res. Lett. 1, No. 6, 809--822 (1994; Zbl 0853.57019)] of \textit{C. H. Taubes}.