AI 中文总结
该研究在有限正则 CW 复形上引入分量完美的多维离散莫尔斯函数概念，基于严格完美 MDM 函数提出更灵活定义，还展示了在其上限制和扩展 MDM 函数的方法，为复杂拓扑操作奠定基础。
AI 中文摘要
我们在有限正则 CW 复形上引入了分量完美的多维离散莫尔斯（MDM）函数的概念。经典意义上的完美离散莫尔斯函数要求指标为\(p\)的临界胞腔数量恰好等于第\(p\)个贝蒂数，而多参数框架需要拓扑动力学视角，其中临界胞腔自然地聚集成连通分量。基于现有的严格完美 MDM 函数概念，我们提出了更灵活的分量完美定义。我们还展示了如何在有限正则 CW 复形上限制和扩展 MDM 函数，为更复杂的拓扑操作奠定基础。
英文摘要
We introduce the concept of component-perfect multidimensional discrete Morse (MDM) functions on finite regular CW complexes. While perfect discrete Morse functions in the classical sense require the number of critical cells of index $p$ to exactly equal the $p$-th Betti number, the multiparameter framework necessitates a topological dynamics perspective, where critical cells naturally cluster into connected components. Building upon the existing notion of strictly perfect MDM functions, we propose the more flexible definition of component-perfection. We also show how to restrict and extend MDM functions on finite regular CW complexes, laying the groundwork for more complex topological operations.
Comments21 pages, 3 figures. Any comments are welcome
