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<p>We study a critical-point graph as a topological invariant of an isolated critical point of a smooth function on a 3-manifold. We construct a distinguishing graph, which is a complete topological invariant of functions with three critical points on a closed 3-manifold. It specifies the partition of a closed 3-manifold into three 3-dimensional disks. We establish the criteria of topological equivalence and prove the realization theorem. The list of all possible distinguishing graphs whose complexity does not exceed four is presented.</p>

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Distinguishing Graph of a Function with Three Critical Points on a Closed 3-Manifold

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Oleksandr Prishlyak,
  • Volodymyr Kiosak,
  • Oleksandr Savchenko

摘要

We study a critical-point graph as a topological invariant of an isolated critical point of a smooth function on a 3-manifold. We construct a distinguishing graph, which is a complete topological invariant of functions with three critical points on a closed 3-manifold. It specifies the partition of a closed 3-manifold into three 3-dimensional disks. We establish the criteria of topological equivalence and prove the realization theorem. The list of all possible distinguishing graphs whose complexity does not exceed four is presented.