论文标题
在具有两个临界值的平滑函数上
On smooth functions with two critical values
论文作者
论文摘要
我们证明，每个平滑的封闭歧管都接受一个平稳的实价函数，只有两个临界值。我们称此类型A \ Emph {Reeb函数}的函数。我们证明，对于REEB函数，我们可以在此集合是歧管的PL子复合物后立即开处方的最小值（或Maxima）。与Reeb的Sphere定理类似，我们使用此类功能来研究基础歧管的拓扑。在尺寸$ 3 $中，我们给出了具有某些REEB函数的属性$ g $ heegaard分裂的歧管的特征。类似的结果在尺寸$ n \ geq 5 $中得到了证明。
We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a \emph{Reeb function}. We prove that for a Reeb function we can prescribe the set of minima (or maxima), as soon as this set is a PL subcomplex of the manifold. In analogy with Reeb's Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension $3$, we give a characterization of manifolds having a Heegaard splitting of genus $g$ in terms of the existence of certain Reeb functions. Similar results are proved in dimension $n\geq 5$.
