Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a (simplicial, polyhedral or cellular) complex without closed orbits, where each cell may either have no arrows, receive a single arrow from one of its facets, or conversely, send a single arrow into a cell of which it is a facet. By following arrows, one can then construct a natural Floer-type boundary operator. Here, we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows (but not both), again in the absence of closed orbits. Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.

中文翻译：

---

广义离散Morse-Floer理论

Forman开发了一种离散的摩尔斯理论，可以用没有闭合轨道的（简单，多面体或细胞）复合体上的箭头图案来理解，其中每个单元要么没有箭头，要么从其一个面接收单个箭头，或者相反，将单个箭头发送到作为小平面的单元格中。通过遵循箭头，可以构造一个自然的Floer型边界算子。在这里，我们为箭头模式开发了这样一种构造，其中每个像元又可以在没有封闭轨道的情况下支持多个向外或向内的箭头（但不能同时支持两个箭头）。我们的主要技术成就是构造一个边界算子，该算子与0平方成正比，因此可以恢复基础复合物的同源性。