Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of $S^2\times S^2$

RESUMO

We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $\Sigma_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $\Sigma_3$ and $\Sigma_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.