On smooth functions with two critical values
arxiv(2022)
摘要
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 $n\geq 5$.
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关键词
smooth functions
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