Topological complexity of monotone symplectic manifolds

TL;DR

This work determines Farber's topological complexity $\mathsf{TC}$ for closed 4‑dimensional monotone symplectic manifolds with Kodaira dimension not $-\infty$. Using TC‑weight and category‑weight techniques, it shows that toroidally monotone 4‑manifolds satisfy $\mathsf{TC}(M)=9$, while 4‑dimensional spherically monotone manifolds with suitable fundamental group conditions satisfy $\mathsf{TC}(M)\in\{8,9\}$; in the same non‑$-\infty$ setting, these manifolds have Lusternik–Schnirelmann category $\mathsf{cat}(M)=5$. The proofs hinge on the atoroidal/aspherical decomposition of $[\omega]-\lambda c_1$, nonvanishing of $([\omega]-\lambda c_1)^n$, and the corresponding lower bounds via TC‑weight, combined with standard upper bounds. The results extend TC calculations beyond aspherical cases and connect symplectic monotonicity with computational invariants of motion planning, offering precise TC classifications in a new geometric regime.

Abstract

We study Farber's topological complexity for monotone symplectic manifolds. More precisely, we estimate the topological complexity of 4-dimensional spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

Theorems & Definitions (32)

• Theorem 1.1: FTY03
• Theorem 1.2: GM20
• Theorem 1.3: FM20
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8: RO99
• Theorem 1.9
• Definition 2.1