Braided Multisections and Akhmedov's Fake Quadrics

Abstract

We give handlebody decompositions of the complements of the cyclic multisections in $Σ_g\times S^2$ constructed in~\cite{AB}. The 2-handle framings are induced by the product coordinates. We compute the integral homology of the complements from their handle chain complexes, including the torsion subgroup of $H_2$. For the degree pairs $(3,3)$ and $(2,4)$, we specify surface diffeomorphisms inducing the maps on first homology prescribed in~\cite{AY} and recover the integral first homology groups $(\Z/3)^2\oplus\Z/(3m)$ and $(\Z/2)^2\oplus\Z/(4m)$, respectively, for $m\geq1$. For the symmetric $m=1$ sum, we also give a simultaneous surface diagram of the relative product attachments and identify their product framings with the surface framings. The resulting symplectic sums are aspherical and symplectically minimal, and have the rational cohomology ring of $S^2\times S^2$.