Motivated by questions about simplification and topology optimization, we take a discrete approach toward the dependency of topology simplifying operations and the reachability of perfect Morse functions. Representing the function by a filter on a Lefschetz complex, and its (non-essential) topological features by the pairing of its cells via persistence, we simplify using combinatorially defined cancellations. The main new concept is the depth poset on these pairs, whose linear extensions are schedules of cancellations that trim the Lefschetz complex to its essential homology. One such linear extensions is the cancellation of the pairs in the order of their persistence. An algorithm that constructs the depth poset in two passes of standard matrix reduction is given and proven correct.

翻译：受简化和拓扑优化问题的启发，我们采用离散方法研究拓扑简化操作的依赖性与完美Morse函数可达性之间的关系。通过Lefschetz复形上的滤子表示函数，并借助持续性理论将细胞的配对关系表征其（非本质）拓扑特征，我们采用组合定义的消去操作进行简化。核心新概念是这些配对上的深度偏序集，其线性扩展构成了消去调度方案，可将Lefschetz复形修剪至其本质同调。其中一种线性扩展是按持续性顺序消去配对的方案。本文给出了一种在标准矩阵归约两轮遍历中构建深度偏序集的算法，并证明了其正确性。