Topological Complexity of symplectic CW-complexes

TL;DR

The paper generalizes the Grant–Mescher result by proving that a connected $2n$-dimensional CW-complex $X$ carrying an atoroidal cohomology class $u\in H^2(X;F)$ with $u^n\neq0$ satisfies $\mathrm{TC}(X)=4n$, and provides a variant with $n$ classes $u_1,\dots,u_n$ with $u_j^2=0$ and nontrivial cup product. The authors develop TC-weight via the fiberwise join $P_2X$ and employ a Mayer-Vietoris analysis to show the zero-divisor $\overline u=1\times u-u\times1$ has weight 2, enabling a sharp lower bound that matches the natural upper bound. A key technical contribution is a corrected Mayer-Vietoris argument and a singular-cohomology–based framework that works for arbitrary coefficient rings. The results yield new TC computations for products of 3-manifolds and for various group-presentation complexes, broadening the scope of topological complexity beyond smooth manifolds and de Rham cohomology.

Abstract

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.

Theorems & Definitions (30)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 3.1