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：本文通过计算紧致光滑流形上闭轨道（包括同宿轨道与周期轨道）与孤立临界点之间适当流线的数量，为广义Morse-Smale动力系统构建了Floer型边界算子。该原理同样适用于Forman定义的CW复形上一般组合向量场的离散情形。由此，我们可以直接从向量场的流线（V-路径）中恢复出光滑结构与离散结构的$\mathbb{Z}_2$同调群。