In this paper, we develop the notion of a Morse sequence, which provides an alternative approach to discrete Morse theory, and which is both simple and effective. A Morse sequence on a finite simplicial complex is a sequence composed solely of two elementary operations, that is, expansions (the inverse of a collapse), and fillings (the inverse of a perforation). In a dual manner, a Morse sequence may be obtained by considering only collapses and perforations. Such a sequence is another way to represent the gradient vector field of an arbitrary discrete Morse function. To each Morse sequence, we assign a reference map and an extension map. A reference map associates a set of critical simplexes to each simplex of a given complex, and an extension map associates a set of simplexes to each critical simplex. By considering the boundary of each critical simplex, we obtain a chain complex from these maps, which corresponds precisely to the Morse complex. We show that, when restricted to homology, an extension map is the inverse of a reference map. Also we show that these two maps allow us to recover directly the isomorphism theorem between the homology of an object and the homology of its Morse complex. At last, we introduce the notion of a flow complex, which is based solely on extension maps. We prove that this notion is equivalent to the classical one based on gradient flows.

翻译：本文提出了莫尔斯序列的概念，为离散莫尔斯理论提供了一种兼具简洁性与有效性的替代研究路径。在有限单纯复形上，莫尔斯序列仅由两种基本操作构成：扩张（即塌缩的逆运算）与填充（即穿孔的逆运算）。对偶地，该序列亦可仅通过塌缩与穿孔操作生成。此类序列是任意离散莫尔斯函数梯度向量场的另一种表征方式。针对每个莫尔斯序列，我们定义了参考映射与扩展映射：参考映射将给定复形中每个单纯形与一组临界单纯形相关联，而扩展映射则将每个临界单纯形与一组单纯形相关联。通过考察各临界单纯形的边界，由这些映射可导出链复形，该复形精确对应于莫尔斯复形。我们证明在局限于同调范畴时，扩展映射即为参考映射的逆映射。同时，这两类映射使我们能够直接推导出对象同调与其莫尔斯复形同调间的同构定理。最后，我们引入基于扩展映射的流复形概念，并证明该概念与基于梯度流的经典定义具有等价性。