论文标题
离散到连续扩展：lovász扩展和摩尔斯理论
Discrete-to-Continuous Extensions: Lovász extension and Morse theory
论文作者
论文摘要
这是一系列论文中的第一篇，它们在离散数学中的构造与相应的连续类似物中开发系统的桥梁。在本文中，我们在简单的复杂性上建立了福尔曼的离散摩尔斯理论与通过lovász扩展的相关顺序复杂复杂的连续摩尔斯理论（在任何已知的非平滑摩尔斯理论的意义上）之间的等效性。此外，我们在抽象的简单络合物上提出了一个新版本的Lusternik-Schnirelman类别，以弥合经典的Lusternik-Schnirelman定理及其在有限复合物上的离散类似物。更普遍地，我们可以通过采用分段线性（PL）Morse理论和Lovász扩展来提出有关超图的离散理论，希望为探索超图的结构提供新的工具。
This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman's discrete Morse theory on a simplicial complex and the continuous Morse theory (in the sense of any known non-smooth Morse theory) on the associated order complex via the Lovász extension. Furthermore, we propose a new version of the Lusternik-Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik-Schnirelman theorem and its discrete analog on finite complexes. More generally, we can suggest a discrete Morse theory on hypergraphs by employing piecewise-linear (PL) Morse theory and Lovász extension, hoping to provide new tools for exploring the structure of hypergraphs.
