Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds

Abstract

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\ge 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of ${\#}_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\le 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer--Vietoris presentation, and a regular cyclic cover of degree $\mathrm{lcm}(p,q)$. The boundary subgroup ${\pi }_1({\Sigma }_G)\times \mathbb{Z}$ embeds in the fundamental group; this gives a nontrivial Bass--Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

29 pages