AI 中文总结
研究单调辛流形的法伯拓扑复杂度，通过证明特定条件下闭球单调辛流形的性质得出部分流形最大拓扑复杂度为9，还计算了\(S^2\) - 丛爆破的拓扑复杂度等，确定了闭辛4维流形在不同情况的\((\cat(M),\TC(M))\)值。
AI 中文摘要
我们继续对文献[Or25]中开始的单调辛流形的法伯拓扑复杂度进行研究。首先，我们证明一个基本群不包含同构于\(\ZZ\oplus\ZZ\)子群的闭球单调辛流形自动是环面单调的，且单调性常数相同。因此，每个柯达里维度不为\(-\infty\)且基本群不包含\(\ZZ\oplus\ZZ\)（例如，格罗莫夫双曲）的闭4维球单调辛流形具有最大拓扑复杂度\(\TC(M)=9\)。其次，我们计算了亏格\(g\geq2\)的闭可定向曲面上\(S^2\) - 丛所有爆破的拓扑复杂度和柳斯捷尔尼克 - 施尼雷尔曼范畴。特别地，第一个结果中关于柯达里维度的假设不能去掉，闭辛4维流形在本文考虑的三种情况下实现了\((\cat(M),\TC(M))=({\color{red}3},5)\)，\(({\color{red}4},7)\)，\(({\color{red}5},9)\)。
英文摘要
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.
Comments14 pages
