AI 中文总结
该研究引入并研究了参数化拓扑复杂性的幺半与对称化版本，构建了相关理论的参数化类似物，证明两类对称化参数化拓扑复杂性概念一致，并对Fadell–Neuwirth纤维化计算了其不变量。
AI 中文摘要
我们引入并研究参数化拓扑复杂性的幺半（monoidal）与对称化（symmetrized）版本。首先，我们构建了Iwase–Sakai、Aguilar-Guzmán–González（基于Fadell–Husseini方法）及Dranishnikov的幺半拓扑复杂性理论的参数化类似物，研究了参数化Iwase–Sakai猜想并给出其成立的充分条件。随后，我们引入对称化参数化拓扑复杂性，结合幺半视角定义了幺半对称化参数化拓扑复杂性，证明所得概念一致，最后对Fadell–Neuwirth纤维化计算了这些不变量。
英文摘要
We introduce and study monoidal and symmetrized versions of parametrized topological complexity. First, we develop parametrized analogues of the monoidal topological complexity theories of Iwase--Sakai, Aguilar-Guzmán--González (based on the Fadell--Husseini approach), and Dranishnikov. We investigate the parametrized Iwase--Sakai conjecture and provide sufficient conditions under which it holds. We then introduce symmetrized parametrized topological complexity and combine it with the monoidal perspective to define monoidal symmetrized parametrized topological complexity, showing that the resulting notions agree. Finally, we compute these invariants for Fadell--Neuwirth fibrations.
Comments36 pages. Added Section 5.2, which establishes an upper bound for monoidal symmetrized parametrized topological complexity
