Topological complexity of symplectic manifolds

TL;DR

This paper establishes that the topological complexity of symplectically atoroidal manifolds equals twice their dimension, extending known results and providing new calculations for various classes of symplectic manifolds.

Contribution

It proves a fundamental relation between topological complexity and dimension for symplectically atoroidal manifolds, generalizing previous results for symplectically aspherical cases.

Findings

Topological complexity equals twice the dimension for symplectically atoroidal manifolds.

Includes calculations for iterated surface bundles and manifolds with hyperbolic fundamental groups.

Results apply to cohomologically symplectic manifolds.

Abstract

We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the Lusternik--Schnirelmann category of a symplectically aspherical manifold equals its dimension. Symplectically hyperbolic manifolds are symplectically atoroidal, as are symplectically aspherical manifolds whose fundamental group does not contain free abelian subgroups of rank two. Thus we obtain many new calculations of topological complexity, including iterated surface bundles and symplectically aspherical manifolds with hyperbolic fundamental groups. Our result also applies in the greater generality of cohomologically symplectic manifolds.