Homeomorphism type of nonstandard symplectic cohomology S^2 x S^2 manifolds

Determine the homeomorphism type of the nonstandard spin symplectic four-manifolds with the same integer cohomology ring as S^2 x S^2 constructed in earlier works (for example, the constructions reported by Akhmedov–Park and by Fintushel–Stern–Park).

Background

Earlier work constructed symplectic four-manifolds that share the integer cohomology ring of S2 x S2 but are not diffeomorphic to S2 x S2. These examples are believed to be not simply connected, yet their precise topological classification has not been established.

This paper constructs related manifolds and demonstrates new phenomena, but explicitly notes that the homeomorphism type of the previously known examples has not been determined. Clarifying the homeomorphism type would resolve the fundamental topological status of these symplectic cohomology S2 x S2 manifolds.

References

Their examples are presumably not simply connected, in particular their homeomorphism type remains unknown.

— Distinguishing closed 4-manifolds by slicing (2505.14387 - Lidman et al., 20 May 2025) in Introduction, paragraph discussing [AkhmedovSymplectic, FSP] (preceding Theorem 1+1)