Odd connected-sum rational cohomology cases

Classify the integers $p,q\geq2$ and genera $g,h$ for which a mixed sum has the rational cohomology ring of $k(S^2\times S^2)$ for an odd integer $k\geq3$, and classify all such quadruples $(p,q,g,h)$ for each odd $k\leq15$.

Background

The paper analyzes mixed sums of cyclic multisections in ruled products and classifies the cases with Euler characteristic $4$, yielding rational cohomology equivalent to S2×S2S^2\times S^2. It develops general formulas for the Euler characteristic, signature, and rational first homology in terms of the covering degrees and base genera.

The authors explicitly ask for a broader geography classification in which the resulting mixed sums have the rational cohomology ring of an odd connected sum of copies of S2×S2S^2\times S^2, rather than only the k=1k=1 case treated in the paper.

References

More generally, for which integers $p,q\geq 2$ and genera $g,h$ does a mixed sum have the rational cohomology ring of

k(S2\times S2)

for some odd $k\geq 3$? Classify all possible quadruples $(p,q,g,h)$ for each such odd $k \leq 15$?

— Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of $S^2\times S^2$  (2609.11727 - Akhmedov, 10 Sep 2026) in Section 'Some questions', second Question