Maximal topological complexity of monotone symplectic 4-manifolds

Ryuma Orita
arXiv:2607.20886

Overview
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Farber’s topological complexity measures how many continuous rules are needed for motion planning on a space. This paper gives conditions ensuring that it reaches the maximum value $\mathrm{TC}(M)=9$ for closed symplectic four-manifolds. Building on our earlier work on monotone symplectic manifolds, we settle the remaining choice between $8$ and $9$ under weaker hypotheses.

The key observation is that a closed spherically monotone symplectic manifold is also toroidally monotone, with the same monotonicity constant, if its fundamental group contains no subgroup isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. For such four-manifolds, this gives $\mathrm{TC}(M)=9$ whenever the Kodaira dimension is not $-\infty$. Gromov hyperbolic fundamental groups are among those satisfying the group condition.

We also compute the invariants for all blowups of $S^2$-bundles over closed orientable surfaces of genus $g\geq 2$: they have $\mathrm{cat}(M)=4$ and $\mathrm{TC}(M)=7$, where $\mathrm{cat}$ denotes Lusternik–Schnirelmann category. These examples show why the Kodaira dimension assumption matters. Together, the three regimes considered in the paper realize $(\mathrm{cat}(M),\mathrm{TC}(M))=(3,5)$, $(4,7)$, and $(5,9)$. All values use the unreduced convention.

BibTeX

@misc{orita2026maximal,
  author        = {Orita, Ryuma},
  title         = {{Maximal topological complexity of monotone symplectic 4-manifolds}},
  year          = {2026},
  eprint        = {2607.20886},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2607.20886}
}

Overview#

Overview
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