标题： 强离散 Morse 理论 显示英文标题

标题： Strong discrete Morse theory

摘要： 本工作的目的是为单纯复形开发一种基于内部强坍缩的Forman离散 Morse 理论版本。 经典的离散 Morse 理论可以看作是Whitehead 坍缩的推广，其中每个定义在单纯复形$K$上的 Morse 函数定义了一个基本内部坍缩序列。 这种约简保证了存在一个与$K$同伦等价的 CW 复形，其单元对应于 Morse 函数的临界单形。 然而，这种方法缺乏对附着映射的显式组合描述，这限制了$K$同伦类型的重建。 通过将离散 Morse 函数限制为由顶点上的全序诱导的函数，我们发展了一种强离散 Morse 理论，推广了 Barmak 和 Minian 引入的强坍缩。 我们证明，在这种情况下，得到的约简 CW 复形是正则的，使我们能够以组合方式恢复其同伦类型。 我们还提供了一个计算此约简的算法，并将其应用于 Benedetti 和 Lutz 的三角剖分库中的复形，以获得高效的结构。 显示英文摘要

摘要： The purpose of this work is to develop a version of Forman's discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead's collapses, where each Morse function on a simplicial complex $K$ defines a sequence of elementary internal collapses. This reduction guarantees the existence of a CW-complex that is homotopy equivalent to $K$, with cells corresponding to the critical simplices of the Morse function. However, this approach lacks an explicit combinatorial description of the attaching maps, which limits the reconstruction of the homotopy type of $K$. By restricting discrete Morse functions to those induced by total orders on the vertices, we develop a strong discrete Morse theory, generalizing the strong collapses introduced by Barmak and Minian. We show that, in this setting, the resulting reduced CW-complex is regular, enabling us to recover its homotopy type combinatorially. We also provide an algorithm to compute this reduction and apply it to obtain efficient structures for complexes in the library of triangulations by Benedetti and Lutz.