We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension 4 and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly μ node and ν saddle equilibrium points of indices equal to either 1 or n — 1. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius \(g=\frac{\nu-\mu+2}{2}\). We also suggest an algorithm for constructing gradientlike flows on closed manifolds of dimension 3 and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.

中文翻译：

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具有给定的非漂移集的允许类似梯度流动的流形的拓扑

我们研究了梯度流的平衡点集的结构与尺寸为4或更高的支撑歧管的拓扑之间的关系。我们引入一类流形，它们接受广义Heegaard分裂。我们认为梯度状流动，使得非游荡集包括恰好的μ节点和ν鞍平衡指数的点等于1或ñ - 1.我们表明，对于这样的流动，存在的一般化Heegaard分裂天才\（g = \ frac {\ nu- \ mu + 2} {2} \）的支撑流形。我们还建议了一种算法，该算法可在尺寸为3或更高且具有指定索引的节点数和鞍形平衡点的闭合歧管上构造梯度流。