Maximal topological complexity of monotone symplectic 4-manifolds

Abstract

We continue the study of Farber's topological complexity for monotone symplectic manifolds initiated in \cite{Or25}. First, we show that a closed spherically monotone symplectic manifold whose fundamental group contains no subgroup isomorphic to $\ZZ\oplus\ZZ$ is automatically toroidally monotone, with the same monotonicity constant. As a consequence, every closed $4$-dimensional spherically monotone symplectic manifold whose Kodaira dimension is not $-\infty$ and whose fundamental group contains no $\ZZ\oplus\ZZ$ (for instance, is Gromov hyperbolic) has maximal topological complexity $\TC(M)=9$. This settles, under strictly weaker hypotheses, the dichotomy $\TC(M)\in\{8,9\}$ left open there. Second, we compute the topological complexity and the Lusternik--Schnirelmann category of all blowups of $S^2$-bundles over closed orientable surfaces of genus $g\geq 2$: they satisfy $\cat(M)=4$ and $\TC(M)=7$. In particular, the hypothesis on the Kodaira dimension in the first result cannot be removed, and closed symplectic $4$-manifolds realize the pairs $(\cat(M),\TC(M))=(3,5)$, $(4,7)$, $(5,9)$ in the three regimes considered in this paper. Throughout, $\TC$ and $\cat$ are taken in the unreduced convention.

Disclosure

“ows that it cannot be removed even in the presence of toroidal monotonicity and a hyperbolic fundamental group. Equivalently, in Corollary 1.5 the assumption κ(M, ω) ̸= −∞ is indispensable. Use of AI. The author made substantial use of the large language model Claude (Anthropic; model Claude Fable 5) during the development of this work. In par- ticular, the key ideas of the proofs of Theorems 1.4 and 1.8 — notably the passage from spherical to toroidal monotonicity via Lemma 3.2 — as well as ini”

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2026-08-23T19:02:47.328853Z

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