Topology related to graph complex

我们在圆上的曲面丛闭3-流形M中引入了“类midakuji路”的概念，并将其应用于M的Z-等变不变量的构造。一条类似amidakuji的路径是M中的一条分段光滑路径，可以被认为是曲面丛投影的梯度向量场的积分曲线的近似。作为Kenji福谷的莫尔斯同伦理论的一个类似物，我们通过对M中边为amidakuji-like路的3-度图进行计数，得到了Z-等变不变量的一个候选.

We introduced the notion of ``amidakuji-like path'' in a closed 3-manifold M that is a surface bundle over the circle and applied it to the construction of a Z-equivariant invariant of M. An amidakuji-like path is a piecewise smooth path in M and can be considered as an approximation of an integral curve for the gradient vector field of the projection of the surface bundle. As an analogue of Kenji Fukaya's Morse homotopy theory, we obtained a candidate for a Z-equivariant invariant by counting 3-valent graphs in M whose edges are amidakuji-like paths.