Finite first-homology groups of the product mixed sums

Determine which finite abelian groups occur as the first homology groups of the three product mixed sums associated with degree pairs $(2,3)$, $(2,4)$, and $(3,3)$, and establish whether the lower bounds on their orders are sharp, including whether $b\mathbb Z/6d$ can occur in the $(2,3)$ case.

Background

The paper constructs mixed symplectic fiber sums from cyclic multisections in ruled products. In the three cases with Euler characteristic $4$ and signature $0$, namely (p,q)=(2,3),(2,4),(3,3)(p,q)=(2,3),(2,4),(3,3), suitable product-framed gluings have the rational cohomology ring of S2×S2S^2\times S^2. Their finite first-homology groups can have unbounded order.

For product-framed gluings, the paper proves that the order of the finite first-homology group is divisible by pqpq, giving lower bounds $6$, $8$, and $9$ in the three cases. The authors explicitly leave open the classification of the finite abelian groups that can occur and whether these divisibility bounds are attained.

References

For each of the three cases in Theorem~\ref{thm:classification}, which finite abelian groups occur as $H_1(Z_\phi;\mathbb Z)$? For product-framed gluings Proposition~\ref{prop:productH1} shows that the order is divisible by

pq=6,\quad 8,\quad 9

in the $(2,3)$, $(2,4)$, and $(3,3)$ cases, respectively. Are these lower bounds sharp? In particular, can one realize $\mathbb Z/6$ in the $(2,3)$ case?

— 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', first Question

Can Luttinger surgeries on Lagrangian tori disjoint from the multisections alter the boundary maps so that the residual integral quotients disappear and the final amalgam becomes simply connected?

— 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', third Question