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arXiv · 2003.06021

Discrete-to-Continuous Extensions: Lov\'asz extension and Morse theory

Jürgen Jost·Dong Zhang
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Abstract

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 Lov\'asz 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 Lov\'asz extension, hoping to provide new tools for exploring the structure of hypergraphs.

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Jürgen Jost, Dong Zhang. 2020-03-12. Discrete-to-Continuous Extensions: Lov\'asz extension and Morse theory. https://arxiv.org/abs/2003.06021

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