Topology related to graph complex

我们在一个封闭的3流形M（即圆上的曲面束）中引入了“amidakuji-like path”的概念，并将其应用于M的z等变不变量的构造。一个amidakuji-like path是M中的一个分段光滑路径，可以被认为是曲面束投影的梯度向量场的积分曲线的近似。作为深谷健二莫尔斯同伦理论的类比，我们通过计算M中边为类偕井路径的三价图，得到了一个z等变不变量的候选点。

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.