We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $\mathbb{Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

翻译：我们通过计算封闭(包括同临床和定期)轨道与孤立临界点之间的适当流动线数目,在紧凑的平滑多管线上为一般摩斯-Smal型动态系统建造一个浮质型边界操作员,对Forman在CW综合体上界定的一般组合矢量场的离散情况适用同一原则,因此我们可以从我们矢量场的流线(V-paths)直接回收光滑和离散结构的同质值。