Topological Complexity of symplectic CW-complexes

A cohomology class u of a topological space X is atoroidal if its pullback to the torus vanishes for every map from a torus to X. Furthermore, X is atoroidally symplectic if there is an atoroidal cohomology class $u\in H^2(X;F)$ such that $u^n$ is non-zero. We prove that every atoroidally symplectic CW-complex X of dimension 2n has topological complexity 4n. This generalizes a result of Grant and Mescher who prove the corresponding statement in the case where X is an atoroidally c-symplectic manifold and u is a de Rham cohomology class. Using this generalisation, we obtain new calculations of topological complexity, including for many products of 3-manifolds and of group presentation complexes.