We develop a general method for constructing random manifolds and sub-manifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel–Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.

我们开发了一种构造任意维度的随机流形和子流形的通用方法。该方法基于将颜色与三角流形的顶点相关联，正如 Sheffield 和 Yadin (2014) 最近针对 3 维空间中的曲线所做的工作一样。我们根据 Stiefel-Whitney 类和其他性质来确定子流形出现的条件。然后，我们考虑因对顶点进行随机着色而产生的随机子流形。由于该模型生成子流形，因此它允许研究属性并使用在产生一般随机子复合体的过程中不可用的工具。三角形三球中的 3 种颜色的情况会产生随机的结和链接。在这种情况下，我们回答了 de Crouy-Chanel 和 Simon (2019) 提出的问题，表明生成不结的概率呈指数衰减。在d维流形中k 种颜色的一般情况下，随着三角剖分中顶点数量的增加，我们研究不同余维的随机子流形。我们计算预期的欧拉特征，并讨论同调渗流和其他拓扑性质的关系。最后，我们探索了一种通过生成随机子流形来搜索拓扑问题的解决方案的方法。我们描述了在 4 维球中搜索低亏曲面的计算机实验，其边界是 3 维球体中的给定结。